EXERCISES IN

STYLE

(Mathematical)

John M

cCle

ary

Exercises in (Mathematical) Style

Stories of Binomial Coefficients

10.1090/nml/050

Originally published by

The Mathematical Association of America, 2017.

ISBN: 978-1-4704-4783-0

LCCN: 2017939372

Copyright c© 2017, held by the Amercan Mathematical Society

Printed in the United States of America.

Reprinted by the American Mathematical Society, 2018

The American Mathematical Society retains all rights

except those granted to the United States Government.

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Visit the AMS home page at https://www.ams.org/

10 9 8 7 6 5 4 3 2 23 22 21 20 19 18

Providence, Rhode Island

Exercises in (Mathematical) Style

Stories of Binomial Coefficients

John McCleary

Vassar College

Council on Publications and Communications

Jennifer J. Quinn, Chair

Anneli Lax New Mathematical Library Editorial Board

Karen Saxe, Editor

Timothy G. FeemanJohn H. McCleary

Katharine OttKatherine SochaJames S. Tanton

Jennifer M. Wilson

ANNELI LAX NEW MATHEMATICAL LIBRARY

1. Numbers: Rational and Irrational by Ivan Niven2. What is Calculus About? by W. W. Sawyer3. An Introduction to Inequalities by E. F. Beckenbach and R. Bellman4. Geometric Inequalities by N. D. Kazarinoff5. The Contest Problem Book I Annual High School Mathematics Examinations

1950–1960. Compiled and with solutions by Charles T. Salkind6. The Lore of Large Numbers by P. J. Davis7. Uses of Infinity by Leo Zippin8. Geometric Transformations I by I. M. Yaglom, translated by A. Shields9. Continued Fractions by Carl D. Olds

10. Replaced by NML-3411. Hungarian Problem Books I and II, Based on the Eötvös Competitions12.

o1894–1905 and 1906–1928, translated by E. Rapaport

13. Episodes from the Early History of Mathematics by A. Aaboe14. Groups and Their Graphs by E. Grossman and W. Magnus15. The Mathematics of Choice by Ivan Niven16. From Pythagoras to Einstein by K. O. Friedrichs17. The Contest Problem Book II Annual High School Mathematics Examinations

1961–1965. Compiled and with solutions by Charles T. Salkind18. First Concepts of Topology by W. G. Chinn and N. E. Steenrod19. Geometry Revisited by H. S. M. Coxeter and S. L. Greitzer20. Invitation to Number Theory by Oystein Ore21. Geometric Transformations II by I. M. Yaglom, translated by A. Shields22. Elementary Cryptanalysis by Abraham Sinkov, revised and updated by Todd

Feil23. Ingenuity in Mathematics by Ross Honsberger24. Geometric Transformations III by I. M. Yaglom, translated by A. Shenitzer25. The Contest Problem Book III Annual High School Mathematics Examinations

1966–1972. Compiled and with solutions by C. T. Salkind and J. M. Earl26. Mathematical Methods in Science by George Pólya27. International Mathematical Olympiads—1959–1977. Compiled and with

solutions by S. L. Greitzer28. The Mathematics of Games and Gambling, Second Edition by Edward W.

Packel29. The Contest Problem Book IV Annual High School Mathematics Examinations

1973–1982. Compiled and with solutions by R. A. Artino, A. M. Gaglione,and N. Shell

30. The Role of Mathematics in Science by M. M. Schiffer and L. Bowden31. International Mathematical Olympiads 1978–1985 and forty supplementary

problems. Compiled and with solutions by Murray S. Klamkin32. Riddles of the Sphinx by Martin Gardner33. U.S.A. Mathematical Olympiads 1972–1986. Compiled and with solutions

by Murray S. Klamkin34. Graphs and Their Uses by Oystein Ore. Revised and updated by Robin J.

Wilson

35. Exploring Mathematics with Your Computer by Arthur Engel36. Game Theory and Strategy by Philip D. Straffin, Jr.37. Episodes in Nineteenth and Twentieth Century Euclidean Geometry by Ross

Honsberger38. The Contest Problem Book V American High School Mathematics Examinations

and American Invitational Mathematics Examinations 1983–1988. Compiled andaugmented by George Berzsenyi and Stephen B. Maurer

39. Over and Over Again by Gengzhe Chang and Thomas W. Sederberg40. The Contest Problem Book VI American High School Mathematics Examinations

1989–1994. Compiled and augmented by Leo J. Schneider41. The Geometry of Numbers by C. D. Olds, Anneli Lax, and Giuliana P.

Davidoff42. Hungarian Problem Book III, Based on the Eötvös Competitions 1929–1943,

translated by Andy Liu43. Mathematical Miniatures by Svetoslav Savchev and Titu Andreescu44. Geometric Transformations IV by I. M. Yaglom, translated by A. Shenitzer45. When Life is Linear: from computer graphics to bracketology by Tim

Chartier46. The Riemann Hypothesis: A Million Dollar Problem by Roland van der

Veen and Jan van de Craats47. Portal through Mathematics: Journey to Advanced Thinking by Oleg A.

Ivanov. Translated by Robert G. Burns.48. Exercises in (Mathematical) Style: Stories of Binomial Coefficients by John

McCleary

Other titles in preparation.

Preface

. . . it is extraordinary how fertile in properties the triangle is. Every-one can try his hand.

Treatise on the Arithmetical Triangle, Blaise Pascal

At the age of twenty-one, he wrote a treatise upon the binomialtheorem, which has had a European vogue. On the strength of it hewon the mathematical chair at one of our smaller universities, andhad, to all appearances, a most brilliant career before him. (SherlockHolmes about James Moriarty)

The Final Problem, Sir Arthur Conan Doyle

Who has anything new to say about the binomial theorem at this latedate? At any rate, I am certainly not the man to know. (Moriarty toWatson)

The Seven-Per-Cent Solution, Nicholas Meyer

. . . every style of reasoning introduces a great many novelties, in-cluding new types of (i) objects, (ii) evidence, (iii) sentences, newways of being a candidate for truth or falsehood, (iv) laws, or at anyrate modalities, (v) possibilities. One will also notice, on occasion,new types of classification, and new types of explanations.

Ian Hacking

According to the Oxford English Dictionary, a pastiche is a work that in-corporates several different styles, drawn from a variety of sources, in the styleof someone else. In this book I am consciously imitating the work Exercices deStyle (Éditions Gallimard, Paris, 1947) of Raymond Queneau (1903–1976). Ina page, Queneau introduces a banal story of a man on a bus that he then tells in99 different ways. The manners of speech include philosophical, metaphorical,onomatopoeic, telegraphic; also the forms of an advertisement, a short play, alibretto, a tanka; in passive voice, imperfect tense; and so on. He celebrates thepotential of language, and his experiments in style, even the most demandingones to read, seek to delight the reader with new forms.

vii

viii Preface

Queneau was no stranger to mathematics, having published a note in theComptes Rendus and a paper in the Journal of Combinatorial Theory (bothon s-additive sequences). He also published Cent mille milliards de poèmes(100,000,000,000,000 Poems1), consisting of a set of ten sonnets with a uni-fied rhyme scheme from which a poem is constructed by making a choice ofeach line from among the corresponding lines in the ten sonnets. This schemeproduces the 1014 distinct poems. The poet estimated that it would require morethan 190 million years to recite them all.

Queneau was also the cofounder in 1960 of the experimental writing groupOuLiPo (Ouvroir de Littérature Potentielle). From their earliest meetings theOuLiPo discussed the use of mathematical structures in literary endeavors. Whenstrict rules (constraints) are chosen and followed, a writer needs to find newmodes of expression. By focusing on new operations applied to literary worksor generating new works, writing can become freer and more thorough, a labo-ratory of invention. The canonical example is La Disparation by George Perec(1936–1982), Denoël, Paris, 1969, a novel in which the vowel e has vanished.Such a text is called a lipogram.

Mathematicians and authors well known to the mathematical communityhave been members of the OuLiPo since its origin, including cofounder FrançoisLe Lionnais, Claude Berge, and Michèle Audin.

So what have I tried to do? I have chosen a (not at all) banal object inmathematics—the binomial coefficients—and written 99 short notes in whichthese numbers (or their cousins) play some role. Certain particular propertiesof these numbers appear in different exercises, seen from different viewpoints.The behavior of the binomial coefficients provides a framework that suggestsproperties of related families of numbers. The binomial coefficients even appearas a sort of character in a mathematical story, or they structure the choices ina piece. The temptation to write a mathematical pastiche of Exercices de Stylehas been on my mind for a long time, and it has also been taken up by others:Ludmilla Duchêne and Agnès Leblanc have written the delightful Rationnel monQ (Hermann, Paris, 2010) in which 65 exercises explore the irrationality of thesquare root of 2.

What is style? According to Wikipedia, style is a manner of doing or pre-senting something. In mathematics, there may be an algebraic way to under-stand certain results, and there also may be a combinatorial way to understandthe same fact. Thus viewpoint contributes to style. Other aspects of style mightinclude the method of a proof, be it induction, recursion, or algorithm; or bycontradiction, by the introduction of a fancy zero, or a conscious imitation ofthe proof of a different fact. On the other hand, the presentation of a mathemat-ical proof might take a particular form of discourse.

1 An English translation of Cent mille milliards de poèmes can be found in Oulipo Compendium,compiled by Harry Mathews and Alastair Brotchie. London: Atlas Press; Los Angeles: Make NowPress, 2005.

Preface ix

There is precious little discussion2 of style in mathematics, even if math-ematicians seem to have strong opinions. At the least, mathematicians seek toimitate their most admired authors’ styles of discourse.

Like Moriarty I cannot hope to say anything new about the binomial coef-ficients. I am sure however that something new can be found on some pages bymost readers. The exercises are intended to explore some of the possible waysto communicate mathematical ideas, and to reveal how binomial coefficientsgive mathematicians plenty to talk about. In some of the titles and in the Stylenotes at the end of the book, I try to identify the features of mathematical think-ing, proving, and writing on which each piece is based. I am not immune tothe enthusiasms of the OuLiPo. As a general constraint each piece is no longerthan three pages. The communication is the thing, and I have tried to tell a littlesomething in each exercise.

Mathematics is one of humanity’s oldest and deepest arts. The challenge ofcommunicating mathematical ideas and truths deserves some critical attention,if only for the love of it. Queneau intended his Exercices de Style to, in his words,“act as a kind of rust-remover to literature.” I share the same intentions with himfor my exercises. And, quoting Queneau,

“If I have been able to contribute a little to this, then I am veryproud, especially if I have done it without boring the reader toomuch.”

Raymond Queneau3

How to use this book

The book is too thin to be useful as a doorstop, and it will anchor few pages inthe role of paperweight. I recommend reading bits at random. You will find thatneighboring pages of any particular exercise may be related in topic and so, ifthat topic interests you, by all means explore. Of course, read with a pencil (orpen) at hand to scribble particular cases, missing steps, critical remarks, conjec-tures (the next Fermat?), comments for the author, or shopping lists, as the spiritmoves you. If a fog of obfuscation surrounds any page, acquaint yourself withthe Style notes found at the end of the exercises, containing helpful citations,some of the author’s zany intentions, and valuable clues to hidden messages and

2See, for example, Variations du style mathématique by Claude Chevalley, Revue de Méta-physique et de morale 42(1935) 375–384. A more extended discussion may be found in the ar-ticle of Paolo Mancuso on mathematical style for the Stanford Encyclopedia of Philosophy, at thewebsite: ♣❧❛t♦✳st❛♥❢♦r❞✳❡❞✉✴❡♥tr✐❡s✴♠❛t❤❡♠❛t✐❝❛❧✲st②❧❡✴. Also, for a wide ranging dis-cussion of style in mathematics see the notes from the Oberwolfach workshop, “Disciplines andstyles in pure mathematics, 1800–2000”, 28.II–6.III.2010, organized by David Rowe, Klaus Volk-ert, Phillippe Nabonnand, and Volker Remmert, found at ✇✇✇✳♠❢♦✳❞❡✴♦❝❝❛s✐♦♥✴✶✵✵✾.

3Quoted in Many Subtle Channels by Daniel Levin Becker, Harvard University Press, 2012.

x Preface

inside jokes4. Prerequisites for the text are few. Most of the exercises can be readafter a good high school mathematics education, some require an acquaintancewith calculus, and a few are written in the language of undergraduate coursesin analysis, algebra, even topology. I have marked the entries in the table ofcontents with a star � if calculus is used, two stars if complex numbers, linearalgebra or number theory is expected, and three stars if exposure to higher levelcourses might be helpful. Dear Reader, the styles of thinking and speaking inmore advanced fields of mathematics are fascinating as well, and exercises atthis level will give you a glimpse of what is ahead. You can check the Stylenotes for introductory texts in the relevant fields.

Enjoy the offerings here in the spirit of play in which they were written.

Acknowledgements

Thanks to Hamish Short who introduced me to Queneau’s Exercices de Styleand whose enthusiasm for it has infected me. Anastasia Stevens, Robert Ro-nan, and Lilian Zhao spent the summer of 2014 helping me find more ways inwhich the binomial coefficients appear in mathematics, providing subject matterfor the exercises. Nancy Fox, Curtis Dozier, and Natalie Friedman shared theirrhetorical expertise with me. Thanks also to folks who read bits, sometimesbig bits, of the manuscript and offered their help in making it better: Ming-Wen An, Cristina Ballantine, Matthias Beck, Jan Cameron, Natalie Frank, JerryFurey, Anthony Graves-McCleary, David Guichard, Liam Hunt, Beth Malm-skog, Clemency Montelle, and Kim Plofker. The difficult work of the copy-editor was expertly done by Underwood Dudley whose prose I admire andwhose advice improved this book. All remaining errors are mine. The edito-rial board of the MAA’s Anneli Lax New Mathematical Library series, TimothyG. Feeman, Katharine Ott, Katherine Socha, James S. Tanton, and Jennifer M.Wilson, led by the ever-encouraging Karen Saxe, provided considerable wisdomand close readings of various versions of the manuscript. My thanks to all of you.The book is better for your time and effort. The efforts of Bev Ruedi to meet mytypesetting demands were most appreciated. Thanks also to Steve Kennedy, theMAA’s Acquisition Editor, who immediately got what I was trying to do.

4This sentence is a pangram—a sentence containing all of the letters of the alphabet.

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Combinatorial thinking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

An algebraic relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Combinatorial consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Downtown Carré . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Proof without words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

A geometric representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12�63

�tetrads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

A footnote . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Fancy evaluations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Careful choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

An explicit formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Explicit counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

The diagonal club . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

The Chu-Vandermonde identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Up and down . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Mind the gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Recurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Rabbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Repetition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

xi

xii Contents

Left out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

The Ramsey game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

The Principle of Inclusion and Exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Derangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Seating the visiting dignitaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Multinomials, passively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Don’t choose, distribute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Tanka** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Ode to a little theorem** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Divisibility by a prime** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

A far finer gambit** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Congruence modulo a prime** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Alchemy** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Close reading** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

An algorithm to recognize primes** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Shifting entries** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

DIY primes** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Polynomial relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

The generalized binomial theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Four false starts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Protasis-apodosis** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Matrices** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Bourbaki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Generating repetitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Dialogue concerning generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Counting trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

q-analogues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Contents xiii

A breakthrough** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Partitions of numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Take it to the limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

q-binomial theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

The quantum plane*** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Scrapsheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Mathematical Idol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Sums of powers* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Parts of proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Rings and ideals*** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Mathmärchen*** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

The binomial distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Wormhole points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

You can’t always get what you want . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Matt Hu, Graduate student . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Seeking successes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

TEXNO…AI�NIA* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

An area computation* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Experimental mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Telescopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

The Rechner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Lipogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

18th century machinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Recipe* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Math talk* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Averages and estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Equality*** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

� , by parts* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

xiv Contents

A macaronic sonnet* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

A hidden integral* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Letter to a princess* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Explicit Eulerian numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

Mutperation satticsits*** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Review* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Plus C* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

At the carnival** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Indicators** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

Tweets** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

Beautiful numbers** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

Reminiscences*** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

Winter journal** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

Cellular automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

Tiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

Lattice points* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

Afterlife* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

Matt Hu and the Euler caper*** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

Hypergeometric musings*** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

On the bus*** . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

Style notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

Style notes

Notations. Notation is an important aspect of good mathematics. The other no-tations for the binomial coefficients, such as C.n; k/, C n

k, or nCk , are clut-

tered and they can lead to misinterpretations and obfuscation. The historyof mathematical notation is a fascinating topic. Famous accounts of nota-tion can be found in Florian Cajori’s A History of Mathematical Notations,in two volumes (1928–1929), Open Court Press, and the recent book ofJoseph Mazur, Enlightening Symbols: A Short History of Mathematical No-tation and Its Hidden Powers, Princeton University Press, 2014.

Combinatorial thinking. This exercise presents an alternative way to think aboutbinomial coefficients. We will see in other exercises how relations amongbinomial coefficients can be established by counting the choices of an un-ordered subset of a finite set. This exercise also offers a problem solvingstrategy: rather than tackling a large problem directly, consider smaller ver-sions of the problem first because they may be more manageable and theevident shape of a solution may emerge.

An algebraic relation. If the binomial coefficients are associated to binomials(they are), then the algebraic properties of binomials can lead to relationsamong their coefficients. The relation of the theorem is named for Pascalwho saw how fundamental is it. However, it was known long before Pas-cal’s time in several different cultures. (See the exercise History.) Algebraic

relations will lead to further relations among the�n

k

�. Summation notation

is explained in the exercise. The proof is an example of algebraic thinking.

Combinatorial consequences. At the base of combinatorial arguments there isthe rule of sum: The cardinality of the union of two disjoints sets is thesum of the cardinalities of each set; and the multiplicative principle. If thenumber of ways to doA is n and to doB ism, then the number of ways to doboth A and B is nm. The rule of sum is applied in the proof of the theorem,and the multiplicative principle is applied in the proof of the proposition.

Downtown Carré. The counting of paths in a lattice is a useful representation ofcertain combinatorial problems. In this exercise, the binomial coefficientsarise naturally from the constraints on paths. The name Edwin is a nod to

239

240 Style notes

Edwin Abbott Abbott, the author of Flatland: A Romance in Many Dimen-sions, Dover Publications, Mineola, NY, 1992. Abbott’s main character is“A Square.” The argument here is a combination of previous two exercises.

Proof without words. A proof without words is most successful when the viewerneeds to spend a little time seeing how the proof emerges. To give you somewords to understand this exercise, the first picture shows the arrangementof the 2-element subsets of f1; 2; : : : ; n C 1g into a triangle in a system-atic manner from which the identity is proved. The numbers introduced arethe triangular numbers Tn. In the case of the pyramidal numbers Pn, eachlayer of the pyramid is a triangular number and the three-element subsets off1; 2; : : : ; nC 2g are pictured. The study of families of numbers associatedwith shapes, figurative numbers, is ancient and signals the beginnings ofnumber theory.

A geometric representation. The representation of .a C b/n for the exponent 2

is found in Euclid, Book II, Proposition 4. This appearance of�2

1

�D 2 is

sometimes cited as the first instance of a case of the binomial theorem (seeJ. L. Coolidge, The story of the binomial theorem, Amer. Math. Monthly56(1949), 147–157).

�6

3

�tetrads. Label the C major scale C D 1,D D 2, . . . , B D 7. Then a choice

of three values from f2; 3; : : : ; 7g, together with C, forms a tetrad, four tones

sounded together. There are�6

3

�possible tetrads. All of them occur exactly

once in some spelling in this piece, which, in the first two lines, is a canonat distance one on the familiar Dies Irae.

A footnote. The letters, Epistola prior and Epistola posterior (see Correspon-dence of Isaac Newton, edited by H.W. Turnbull, Cambridge UniversityPress, 1977, volume 2), were written long after Newton had deduced thegeneral binomial theorem for rational exponents (see the exercise Experi-mental mathematics). In response to the curiosity of Leibniz, he describedthe manner in which he discovered the general theorem, giving enough hintsto the power of his methods, and the scope of their application. Leibniz haddiscovered many of the same results in the Epistolae independently. New-ton’s observation about the powers of 11 is truly a footnote to the rest of theconcepts discussed in these famous letters. Other uses of the data providedin Pascal’s triangle are possible along the lines suggested in the footnote;for example, powers of 12. The reader is encouraged to find their own co-incidences.

Fancy evaluations. Algebraic expressions represent functions that may be eval-uated for special values of the variables. The reader can derive an infinite

number of delightful relations in this manner. For example,Xn

kD0

�nk

�2k D

Style notes 241

3n. Polynomials represent a deeper notion, however. They can be under-stood as formal sums that are determined by their coefficients. In this frame-work, two polynomials are equal when their coefficients are equal. Thisprinciple leads us to compare like coefficients for polynomials that are equalfor different reasons. For a discussion of the algebra of polynomials see, forexample, Serge Lang’s book Undergraduate Algebra, Springer-Verlag, 3rded. 2005.

Careful choices. In this exercise we prove by combinatorial means the same re-lations found in the exercise Fancy evaluations. The combinatorial methodsets up a correspondence between the collection of things to be countedwith one property and another collection of things with another property.When the correspondence is perfect, that is, a one-one correspondence, asin a subset and its complement, then the numbers of things in each collec-tion are seen to be the same. Such proofs are called bijective proofs. Bi-jective proofs are sometimes thought to be more “satisfying” as argumentsthan algebraic manipulations. Compare the exercises Careful choices and

Fancy evaluations and decide what you prefer. The relation�n

k

��n � kp

�D

� np

��n � pk

�is sometimes called the subset-of-a-subset identity.

An explicit formula. The absorption identity is a consequence of the evaluationof another identity, the subset-of-a-subset identity. In this exercise we ex-plore how general relations can be evaluated to give particular cases thatmay open up new relations. Explicit formulas are not immediately availablewhen numbers are defined algebraically or combinatorially. Arguments likethe one found in this exercise are often needed to discover explicit formulas.

Explicit counting. The relationship between ordered choices and unordered sub-sets leads to the explicit formula for binomial coefficients. The multiplica-tive principle plays a key role. To test your understanding, work out theordering of the numbers of significant poker hands (pair, two pair, three ofa kind, straight, flush, full house, four of a kind, straight flush): the fewerthe instances, the better the hand.

History. My historical remarks are based on the account found in The Historyof Mathematics by David Burton, 3rd edition, McGraw-Hill Companies,Inc., New York, NY, 1997. The reference to the table of al-Samaw’al comesfrom William Casselman, Binomial coefficients in Al-Bahir fı Al-Jabr, No-tices of the Amer. Math. Soc., 60(2013), 1498. My thanks to Jeff Suzuki,Clemency Montelle, and Kim Plofker for references on Indian mathemat-ics. In particular, the account of the work of Pingala is based on the pa-per Sanskrit Prosody, Pingala Sutras and binary arithmetic by RamaiyengarSridharan, in Contributions to the History of Indian Mathematics, editedby Gérard G. Emch, Ramaiyengar Sridharan, and M. D. Srinvas, Hindustan

242 Style notes

Book Agency, 2005, Gurgaon, India.

The diagonal club. This exercise refers to the fictional place found in DowntownCarré. The connection between binary digits and subsets is another way tosum the binomial coefficients. For a set like fa; b; c; dg, a binary number,say 0101, corresponds to the subset fb; dg. Generally, if you allow binaryexpressions to have zeroes in front to get them to length M , then there isa correspondence between subsets of a set with M elements and binarystrings of length M . Counting in this manner was known to Pingala in histreatise on poetic meters from 200 BCE.

The Chu-Vandermonde identity. What appears in Chu Shih-Chieh’s PreciousMirror of the Four Elements is the identity

nX

rD1

r.r C 1/ � � � .r C p � 1/pŠ

� nC 1 � p/.nC 2 � p/ � � � .nC q � p/qŠ

DnX

rD1

r.r C 1/ � � � .r C p C q � 1/.p C q/Š

;

which can be massaged into a version of the theorem. Joseph Needham(1900–1995), in his classic Science and Civilisation in China, CambridgeUniversity Press, Cambridge, 1954, states that the identity is obtained ina discussion of series and progressions. Vandermonde proved the identityin his Mémoire sur des irrationnelles de différents ordres avec une applica-tion au cercle (1772). His paper treats in great detail the properties of fallingfactorials, .x/k D x.x � 1/ � � � .x � k C 1/. The identity appears on p. 492of his Mémoire. Vandermonde was a medical doctor, violinist, and begancontributing to mathematics at age 35. He is best known for his work ondeterminants. His arguments are developed further in the exercise 18th cen-tury machinations. Three proofs of the identity are found in this exercise:the first algebraic, the second combinatorial, and the third based on latticepath counts.

Up and down. Limericks are a form of light poetry, usually based on the theanapest rhythm (two unaccented syllables followed by an accented one),and usually having three stressed syllables in lines 1, 2, and 5, and twostressed syllables in lines 3 and 4. The rhyme scheme is always AABBA.Limericks of a mathematical sort abound. My favorite ones refer to theMöbius band. One of the best I discovered is by Leigh Mercer and appearedin Word Ways, 13(1980), 36. Mathematically it is:

12C 144C 20C 3p4

7C .5 � 11/ D 92 C 0:

As a limerick it is:

Style notes 243

A dozen, a gross, and a score,Plus three times the square root of four,

Divided by sevenPlus five times eleven

Is nine squared and not a bit more.

A function f .x/ is unimodal if there is a value m for which f .x/ isincreasing when x � m, and decreasing for x � m. Hence f .m/ is amaximum value of f .x/. The unimodal property of binomial coefficients isshared by many other important combinatorial sequences. For a given func-tion or sequence, proving that it is unimodal can be difficult. Unimodalityfor distributions plays a role in probability theory.

Mind the gaps. The general formula for the number of possible distributionsto k individuals (numbered 1 to k) of n identical objects where each in-

dividual gets at least one object is given by

�n � 1k � 1

�. The argument is

exactly how Amalie describes it in the special case. For the number of pos-sible distributions of n identical objects to k individuals where it is possible

for individuals to get nothing, Hermann’s argument gives

�nC k � 1k � 1

�. I

imagine the scene of this exercise in an early 20th century German home,in Göttingen perhaps. Also see the exercise Repetitions.

Recurrence. To define a sequence by a recurrence relation is a kind of axiomaticreduction of the sequence to a simpler form. The binomial coefficients werepresented by Pascal in this form, leading to the arithmetical triangle andmany of the results he obtained. Recurrence relations play an importantrole in computer science, offering, on one hand, a recipe for programminga computation, and on the other hand, straightforward descriptions of se-quences that test the limits of computation.

Rabbits. Both sequences, of binomial coefficients and of Fibonacci numbers,are defined by recurrence relations, where each term is a sum of previousterms. A relation between the two sequences is not unexpected. The title ofthe exercise refers to the original problem posed by Fibonacci (Leonardo ofPisa (c 1170–c. 1250)). There is also a melding of the two notions called

fibonomials: let�nk

�F

D fnfn�1 � � � fn�kC1fkfk�1 � � � f2f1

. The reader might enjoy de-

riving the properties of the fibonomials as modeled by the behavior of bino-

mial coefficients. For example, by proving

�nC 1

k

�

F

D fn�kC2�nk

�F

C

fk�1� n

k � 1�F

, it follows that�nk

�F

is an integer for all n and k. See,

D. K. Hathaway, S. L. Brown, Fibonacci powers and a fascinating triangle,College Math. J. 28(1997), 124–128.

244 Style notes

Repetition. The theorem is another example of an recurrence relation that leadsto an explicit closed form, with the help of Pascal’s identity. The form of therecurrence relation is sufficient for the construction of a table. The inductionproof is more subtle being an induction on a sum of variables rather thana single variable. The odd discourse is inspired by Queneau’s exercise Enpartie double.

Left out. The proof of this identity is a combinatorial argument and employs theordering of the set of elements being counted to make distinctions betweensubsets. This artifact, that you can label the set of N distinct objects bythe ordered labels 1 through N , provides a little extra feature that swingsthe argument. Furthermore, thinking about what is missing often can be aspowerful as thinking about what is there.

Sets. The arguments in this exercise are sometimes called bijective: To count theobjects in a set, find another set whose cardinality is easier to determine andthen construct a bijection between the set you want to count and the set withknown cardinality. The notation emphasizes which sets are involved. If we

remove the assumption that X is finite, it is possible to define�X

Y

�D fA �

X j #A D #Y g. We can ask how the set�X

Y

�and its cardinality behave.

For example, if X D N D Y , the subsequent set is the collection of allcountably infinite subsets of the natural numbers, an uncountable set.

The Ramsey game. The arguments here are based on the well developed and sub-tle study of Ramsey theory, first considered in the paper of Frank P. Ramsey,On a problem of formal logic, Proc. London Math. Soc. 30(1930), 264–286. By recasting the basic theorems of Ramsey theory as a game, playerscan gain a hands-on feel for the combinatorial consequences of these ideas.A clear and thorough exposition of Ramsey theory can be found in RonL. Graham, Bruce L. Rothschild, Joel H. Spencer, Ramsey Theory, Wileyand Sons, New York, NY, 1980. The game is best played on graph paperwith pens of different colors.

The Principle of Inclusion and Exclusion. The principle of inclusion-exclusion isone of the fundamental tools in combinatorics. In the sieve formulation, forexample, to count primes, these ideas were known in antiquity. It is not clearwhen a “first” formal statement of the principle comes into the literature.For the probability version of the principle, it appears in Henri Poincaré’sCalcul des Probabilités, Gauthier-Villars, Paris, 1896. My account is basedon a chapter of the excellent book, Combinatorics: Topics, Techniques, Al-gorithms by P. J. Cameron, Cambridge University Press, Cambridge, UK,1994.

Inversion. Binomial inversion links together certain sequences of numbers in un-expected ways. Applications can be found by expressing a known sequence

Style notes 245

in terms of an unknown sequence and then applying inversion to obtain theunknown sequence. I apply this method in a few of the exercises to come.John Riordan’s excellent book, Combinatorial Identities, Wiley and Sons,New York, NY, 1968, devotes two chapters to inversion. When viewed prop-erly, inversion is an example of the principle of inclusion and exclusion, andeven an example of matrix inversion (see the exercise Matrices).

Derangements. The problem of counting derangements is also known as theproblème des rencontres, and it was first posed in 1708 by Pierre Rémondde Montmort in his Essai d’analyse sur les jeux de hazard. He solved theproblem in 1713 in the third edition of the book. The argument for derange-ments is a beautiful example of the method of inversion, where the knownsequence fnŠg is related by inversion to the unknown sequence fDng. In-version recovers the unknown sequence in terms of the known. By the way,Dn is the integer closest to nŠ=e.

Seating the visiting dignitaries. This combinatorial puzzle is a lightly disguisedversion of the well known problème des ménages, posed in 1891 by ÉdouardLucas in his book Théorie des Nombres, Paris: Gauthier-Villars (pp. 491–495) and solved in the decade following by Charles-Ange Laisant,M. C. Moreau, and H. M. Taylor. The original problem seats alternatinghusbands and wives around a circular table with no husband sitting next tohis wife. I have chosen a non-sexist formulation with the visiting council.A related problem had been considered by Arthur Cayley (1821–1895) in1878 that was shown to be equivalent by Jacques Touchard, Sur un prob-lème de permutations, C. R. Acad. Sciences Paris 198(1934), 631–633. Itis Touchard who gave the formula at the end of the exercise. The proofpresented here follows the classic proof of Irving Kaplansky, Solution ofthe ‘Problème des ménages’, B. Amer. Math. Soc., 49(1943), 784–785. An-other non-sexist version is given by Kenneth Bogart and Peter G. Doyle,Non-sexist solution of the ménage problem, Amer. Math. Monthly 93(1986),514–519.

Multinomials, passively. This exercise features an example of the method of gen-eralization. We have some experience with binomial coefficients, so let’sextend it to more variables. For example, because the binomial coefficientssatisfy the basic identity of Pascal, we can mimic a proof of that relation inthe new context to see if it gives a relation for multinomial coefficients. (Itdoes!) The multinomial theorem was discovered around 1676 by Leibnizbut never published by him. De Moivre published a proof of it indepen-dently 20 years later. (See N. Bourbaki, Elements of the History of Mathe-matics, Springer-Verlag, Berlin, 1998.) If the tone of the exercise seems abit formal, it is because the passive voice is something to be avoided.

246 Style notes

Don’t choose, distribute. The generalized binomial coefficients�nk

�m

were first

studied by Abraham de Moivre (1667–1754) in The Doctrine of Chances:or, A method of calculating the probabilities of events in play (third edition1756, printed for A. Millar, London), the first textbook on probability. DeMoivre analyzed outcomes of games of chance and needed such coefficientsfor counting. Leonhard Euler made a thorough study of these numbers inDe evolutione potestatis polynomialis cuiuscunque .1 C x C x2 C x3 Cx4 C etc./n, Nova Acta Academiae Scientarum Imperialis Petropolitinae12(1801), 47–57. The four results of the exercise are established by a com-binatorial approach. The algebraic results in this exercise are parallel to thecombinatorial arguments in the exercise Four false starts.

Tanka/Haiku. The tanka is a form of Japanese poetry consisting of five lines,with each line containing a restricted number of syllables according to thepattern 5-7-5-7-7. The more familiar haiku has three lines, numbering 5-7-5in syllables. If you are given k consecutive integers, they may be written n,nC 1, . . . , nC k � 1. The product, divided by kŠ is given by

n.nC 1/ � � � .nC k � 1/kŠ

D�nC k � 1

k

�:

Because binomial coefficients are always integers, kŠ divides n.n C 1/ � � �.nC k � 1/. This short proof and the nontrivial nature of the result deservethe interjection Wow!

Ode to a little theorem. For Pierre de Fermat (1601–1665) and his contempo-raries Fermat’s little theorem would be stated: for any integer n and primenumber p, p divides np � n. In the notation of Gauss (1777–1855), intro-duced in his Disquistiones Arithmeticae, one would write np � n.mod p/.If p does not divide n, then np�1 � 1.mod p/. The proof presented herefollows ideas of Fermat; the more familiar proofs emphasize the algebraicproperties of arithmetic modulo p. Induction was developed into a subtletool by the mathematicians of Fermat’s time. The consequences of Fermat’sinsight, Gauss’s notation, and the little theorem are far-reaching in modernelementary number theory (it deserves an ode). My favorite books on ele-mentary number theory include: G. H. Hardy, E. M. Wright, An Introductionto the Theory of Numbers (sixth edition), Oxford University Press, 2008,New York, NY; Trygve Nagell, Introduction to Number Theory, AmericanMathematical Society; 2nd Chelsea Reprint edition (2001); and UnderwoodDudley, Elementary Number Theory, Dover Publications; Second edition(2008), Mineola, NY.

Divisibility by a prime. I based this exercise on the account of Kummer’s the-orem by Paulo Ribenboim in The New Book of Prime Number Records,Springer Verlag, NY, third edition 1996, and the e-survey of Andrew

Style notes 247

Granville, Arithmetic properties of binomial coefficients, at ✇✇✇✳❞♠s✳

✉♠♦♥tr❡❛❧✳❝❛✴�❛♥❞r❡✇✴❇✐♥♦♠✐❛❧✴. Kummer’s result appeared in thepaper Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen,J. reine angew. Math. 44(1852), 93–146, where Kummer analyzed the be-havior of cyclotomic sums in order to prove a generalization of the law ofquadratic reciprocity. The use of p-adic representations of integers is natu-ral in this situation. Also the modest process of carrying in addition takes onconsiderable significance in the analysis of divisibility. Nothing we learn,even from elementary school, is too modest to be useful.

A far finer gambit. The title of this exercise is taken from G.H. Hardy’s A Math-ematician’s Apology, Cambridge University Press, Cambridge, UK, 1940,in which he wrote “The proof is by reductio ad absurdum, . . . one of a math-ematician’s finest weapons. It is a far finer gambit than any chess gambit: achess player may offer the sacrifice of a pawn or even a piece, but a mathe-matician offers the game.” The method of reductio ad absurdum or proof bycontradiction is an important tool for fashioning proofs. The best known ex-amples include Euclid’s proof of the infinitude of primes, and the arithmeticproof of the irrationality of

p2. The study of squarefree binomial coeffi-

cients was initiated by Erdos in Pal Erdos and Ron L. Graham, Old and NewProblems and Results in Combinatorial Number Theory, L’Enseignement

Math. Monographie 28, Genève, 1980. There Erdos conjectured that�2m

m

�

is squarefree for m > 4. This was proved for m > n0 for a large valuen0 by A. Sárkozy, On divisors of binomial coefficients. I., J. Number Th.20(1985), 70–80. A complete proof following Sárkozy’s ideas was givenby A. Granville, and O. Ramaré, Explicit bounds on exponential sums andthe scarcity of squarefree binomial coefficients, Mathematika 43(1996), 73–107. The lemma and theorem of the exercise, and the arguments, are fromtheir paper. The authors sharpen the understanding of squarefree binomialcoefficients, proving that “on average, there are approximately ten-and-two-thirds squarefree entries in a row of Pascal’s triangle.”

Congruence modulo a prime. This application of Fermat’s little theorem is par-ticularly satisfying, revealing the power of modular arithmetic. I chose atwo-column proof for reasons of nostalgia. The results here first appearedin the papers É. Lucas, Sur les congruences des nombres eulériens et lescoefficients différentiels des functions trigonométriques suivant un modulepremier, Bull. de la Soc. Math. de France 6(1878) 49–54; J. Glaisher, Onthe residue of a binomial-theorem coefficient with respect to a prime mod-ulus, Quart. J. of Pure and Applied Math. 30(1899), 150–156; and N. Fine,Binomial coefficients modulo a prime, Amer. Math. Monthly 54(1947) 589–592.

248 Style notes

Alchemy. This exercise is based on the first part of the paper On certain prop-erties of prime numbers Quarterly J. of Pure and Applied Math. 5(1862),35–39) by the Reverend Joseph Wolstenholme (1829–1891). I have takensome liberties in the derivation (for example, a little calculus) of the for-

mula for Hn. Wolstenholme shows how the sumXn

jD1.�1/jC1

�n

j

�1

j

has a first difference, as a function of n, of1

n. This does not suggest how he

came up with the formula. One lesson to learn from this exercise is that sim-plification of expressions can hide potential riches. Throughout the deriva-tion of the divisibility result we have chosen to consider a more compli-cated expression. A motto might be: “Simplify only if it leads to somethingricher.” The reference to Oresme and the harmonic series may be found inQuaestiones super geometriam Euclidis, edited by H.L.L. Busard, Leiden1961; corrected version: J. E. Murdoch, Nicole Oresmes “Quaestiones su-per geometriam Euclidis,” Scripta Mathematica 27(1964), 67–91. I aimedfor an alchemical tone based on readings from The Alchemists by F.S. Tay-lor, Henry Schuman, NY, 1949, especially the passage attributed to NicholasFlammel (pp. 163–173).

Close reading. The exercise focuses on the explicit forms of the coefficientsc1 and c2 whose divisibility properties lead to the divisor of p3. Togetherwith Alchemy, it completes my close reading of Wolstenholme’s 1862 paper.Other proofs of Wolstenholme’s theorem may be found using methods suchas Lucas’s theorem; for example, see J.W.L. Glaisher, On the residues of thesums of products of the first p � 1 numbers, and their powers, to modulusp2, or p3, Quart. J. Math. 31(1900), 321–353. The remarkable review paperof Andrew Granville, Arithmetic properties of binomial coefficients. I. Bi-nomial coefficients modulo prime powers, in Organic Mathematics (Burn-aby, BC, 1995), CMS Conf. Proc., vol. 20, American Mathematical Society,Providence, RI, 1997, 253–275, contains many recent results of Wolsten-holme type. See also R. Mestorovic, Wolstenholme’s theorem: Its gener-alizations and extensions in the last hundred and fifty years (1862–2012),

arXiv:1111.3057v2. These papers include a relation between�2p � 1p � 1

�and

the Bernoulli numbers and the latest news on the search for Wolstenholmeprimes.

An algorithm to recognize primes. One of the most desirable results in math-ematics is a necessary and sufficient condition for a positive integer to beprime. The best known example is Wilson’s theorem

n � 2 is prime if and only if n divides .n � 1/ŠC 1:

Babbage’s paper, Demonstration of a theorem relating to prime numbers,Edinburgh Philos. J. 1(1819), 46–49, contains the theorem on which the

Style notes 249

algorithm is based. It also contains the result that p2 divides�2p � 1p � 1

�if p

is a prime. The Chu-Vandermonde identity plays a useful role in his proofs.

Think about�2p

p

�� 2. The pseudocode brings up the question of finding

an efficient algorithm to compute�n

k

�given n and k. Such an algorithm

would be needed to analyze the efficiency of this primality test.

Shifting entries. The paper of Mann and Shanks is very readable and gives aslightly different argument. In their paper the authors discuss their moti-vation for proving the result. They were studying the product over primesY1

pD2

�1 � 1

p2� 1

p3

�. The idea of Granville mentioned in the proof may

be found in Arithmetic properties of binomial coefficients. I. Binomial co-efficients modulo prime powers, in Organic Mathematics (Burnaby, BC,1995), CMS Conf. Proc., vol. 20, American Mathematical Society, Provi-dence, RI, 1997, 253–275.

DIY primes. I first learned Erdos’s proof of Bertrand’s postulate in Wacław Sier-pinski’s Elementary theory of numbers, translated from Polish by A. Hulan-icki. Warszawa, Panstwowe Wydawn. Naukowe, 1964. The proof is a tour-de-force of elementary ideas, all in the right place. I hope you will dive inand fill in all the details for yourself. My Do-It-Yourself kit is based on thelovely exposition of the proof in Proofs from the BOOK by Martin Aignerand Günter Ziegler, 4th edition, Springer-Verlag, Berlin, 2010. Pál Erdoswas one of the most influential and productive mathematicians of the twen-tieth century. He would speak of the BOOK in which God has collected theperfect proofs of mathematical theorems. In 1932, aged 19, Erdos publishedthe paper Beweis eines Satzes von Tschebyschef, Acta Sci. Math. (Szeged)5(1930-32), 194–198, in which he proved Bertrand’s postulate as outlinedin the exercise—certainly a proof to be found in the BOOK.

Polynomial relations. The main tool of this exercise, the lemma on polynomi-als and their shared values, can be used in ingenious ways to prove manyrelations. This useful fact can be applied, for example, when studying thegeometric properties of polynomial functions, that is, in the study of al-gebraic geometry. Among the generalizations of binomial coefficients, the

polynomials�x

k

�play an important role, interpolating between the integer

values for x. See the exercise 18th century machinations for a developmentof such ideas.

The generalized binomial theorem. It is an attractive aspect of mathematics thatquestions like What if? can be followed on by exploring as if an assertion istrue to find out some of the consequences of a statement before proving it.Famously, Hardy and Littlewood explored the consequences of the failure

250 Style notes

of the Riemann Hypothesis, a failure that they thought would not be a badthing for number theory. The proof in the exercise is based on the proofof Auguste Cauchy (1789–1857) of Newton’s binomial theorem found inhis 1821 Cours d’analyse algébrique. The parts of the proof in the exer-cise pass muster in the present day regard for rigor. His assertion that thefunctions f .r; x/ are continuous because they are sums of continuous func-tions, however, does not meet this standard, as was pointed out by Abel inhis remarkable 1826 paper on the binomial series. See the related exercisesEquality and Afterlife.

Four false starts. The title of this exercise is taken from Forty-one false starts:essays on artists and writers by Janet Malcolm (Farrar, Straus and Giroux,New York, 2013). False starts are a way of life in mathematics. The reasonsfor abandoning a course of development are sometimes computational: Theright expression for a coefficient may depend on what role the numbers are

going to play. These generalized binomial coefficients�nk

�m

were initially

studied in the first textbook on probability by Abraham de Moivre in TheDoctrine of Chances. De Moivre analyzed outcomes of games of chancefor which he needed such coefficients for counting. Euler made a thor-ough study of these numbers in De evolutione potestatis polynomialis cuius-cunque .1C xC x2 C x3 C x4 C etc./n, Nova Acta Academiae ScientarumImperialis Petropolitinae 12(1801), 47–57. An interesting recent preprintby N.-E. Fahssi, Polynomial triangles revisited (arXiv:1202.0228v7) givesseveral interpretations of the Pascal-de Moivre triangle, including connec-tions to physics.

Protasis-apodosis. The grammatical terms protasis and apodosis refer to theantecedent and consequent, respectively, of an if-then sentence. From theGreek, protasis is to stretch before, and apodosis, to give back. Grammati-cal implications can vary with the language. I learned of these terms froma discussion of the omen literature of ancient Babylonia in Clemency Mon-telle, Chasing Shadows: Mathematics, Astronomy, and the Early History ofEclipse Reckoning, John Hopkins U. Press, Baltimore, MD, 2011. Anotherexample of this form of discourse is found in the children’s book, If YouGive a Mouse a Cookie by Laura Joffe Numeroff and Felicia Bond, HarperCollins, New York, 2010. This exercise reveals some of the power of lin-ear algebra to organize spaces of polynomials. The Stirling numbers of thefirst and second kind are introduced and their basic properties derived. Fora fascinating history of the Stirling numbers and more of their properties,see the article of Khristo N. Boyadzhiev, Close encounters with the Stirlingnumbers of the second kind, Math. Mag. 85(2012) 252–266.

Matrices. Viewing the Pascal triangle as a matrix is a natural temptation and itleads to new formulations of the properties of the binomial coefficients. For

Style notes 251

example, the existence of an inverse provides a new look at the method ofbinomial inversion. When studying matrices, certain operators on the ma-trices can be defined, and in this case we find the shift operator and thederivative of Kemeny. The relation P D P 0 is a different statement of thePascal identity, which invites us to consider other compartments of mathe-matics (analysis in this case). Matrix and operator proofs of combinatorialsums may be traced back to work of Leonard Carlitz, On arrays of num-bers, Amer. J. Math. 54(1932), 739–752. More general systems that resultfrom infinite matrices like P have been investigated by others under thename of Riordan arrays (see R. Sprugnoli, Riordan arrays and combina-torial sums, Discrete Math. 132(1994), 267–290), and Galton arrays (seeE. Neuwirth, Recursively defined combinatorial functions: extending Gal-ton’s board, Discrete Math. 239(2001), 33–51.

Bourbaki. Nicholas Bourbaki, the author of the celebrated Éléments de Mathe-matique, is unjustly accused of a dry and pedantic style of exposition. Thistone may be the result of the joint nature of the writing. Bourbaki was agroup of young French mathematicians, who first met together in 1934 withthe goal of producing a new and practical curriculum for analysis in theFrench universities. The project and its history are told in the book of Mau-rice Mashaal, Bourbaki: A Secret Society of Mathematicians. Translated byAnna Pierrehumbert. Providence, RI: Amer. Math. Soc., [2002] 2006. Theanalysis of mappings and the language of surjections, injections, and bijec-tions was developed in their work. Bourbaki never published an account ofles artes combinatoires. The symbol “dangerous bend,” adopted from road-side signs, appears in the works of Bourbaki to indicate difficult aspects ofan argument. The numbers S.n; k/ D #SUR.Œn�; Œk�/=kŠ are, in fact, thesame as the Stirling numbers of the second type that appear in the exerciseProtasis-apodosis. The reader may enjoy exploring the equivalence of thesecounts.

Generating repetitions. Generating functions and the algebra of formal powerseries provide parts of the foundation of the study of combinatorics. Theeconomy of expression, ease of manipulation, and sometimes startling re-lations revealed by generating functions make them a very attractive objectof study. See the classic book by Herb Wilf, generatingfunctionology, 3rdedition, AK Peters, 2006, Wellesley, MA. Questions of convergence of for-mal power series are not considered, only their algebraic structure. Equal-ity in this context is equality of formal power series—like powers haveequal coefficients—an algebraic criterion that does not require evaluationat a value of x to produce an equation.

Dialogue concerning generating functions. The Catalan numbers

Cn D 1

nC 1

�2n

n

�D�2n

n

���2n

n � 1

�

252 Style notes

were introduced by Euler to count the number of ways to divide a poly-gon into triangles. The sequence is named after Eugène Charles Catalan(1814–1894), who found that the values count the number of ways one canintroduce parentheses to a product of (non-associative) variables and ob-tain a meaningful product. The sequence also appears in the work of theChinese mathematician Ming’antu (c.1692–c.1763) on infinite series ex-pressions for sin.2x/ in terms of sin.x/. The ubiquity of Catalan numbersand their properties can be explored in the recent book Catalan Numbers byRichard P. Stanley, Cambridge University Press, New York, 2015. This ex-ercise presents the standard derivation of the values of the Catalan numbers.It illustrates the power of generating functions to obtain explicit formulas,even for complicated recurrences. I particularly like how the quadratic for-mula makes an appearance. The binomial theorem for rational exponentsgives the needed infinite series. The title and discourse are based on Plato’sMeno and on Galileo’s Dialogue concerning the two chief world systems.

Counting trees. To see the relation between Euler’s problem of counting thenumber of ways to triangulate an .n C 2/-gon, Catalan’s problem of par-enthetization and the count of planar, rooted, trivalent trees, consider thediagram

((ab)c)d (a(bc))d (ab)(cd ) a((bc)d ) a(b(cd ))

1

1

2

3

4

51

2

3

4

5 1

2

3

4

5 1

2

3

4

5 1

2

3

4

5

154 2 543 1 53 1 2 54 1 2 53

The trivalent property of our trees may be interpreted for parenthetization asthe two inputs and one output of a multiplication. To get a sense of how theCatalan numbers describe many other phenomena, see Richard Stanley’sbook Catalan Numbers which contains Igor Pak’s paper on the history ofthe Catalan numbers (Appendix B).

q-analogues. The Gaussian polynomials, or q-binomial coefficients,

�n

k

�were

introduced by Gauss in an 1811 paper, in order to study questions about thebehavior of cyclotomic polynomials. This idea and the role of q-binomialcoefficients in the theory of partitions and number theory will be consideredin other exercises. Analogy can be a powerful source of potential results,and here I study the Gaussian polynomials as if I were studying the binomialcoefficients. The use of the lexicographic order for induction is justifiedbecause the initial conditions in each portion of the total order on pairs hold

Style notes 253

from the fact that

�m

0

�D 1 D

�m

m

�and that we can take

�m

mC k

�D 0

for k > 0. Wonderful sources for properties of Gaussian polynomials arethe books of George E. Andrews and Kimmo Eriksson, Integer Partitions,Cambridge University Press, New York, 2004, and George E. Andrews, TheTheory of Partitions, Cambridge University Press, New York, 1998.

A breakthrough. These “notes” constitute a paraphrase, with mathematics, of acelebrated letter of Carl-Friedrich Gauss to Heinrich Olbers (1758–1840)dated September 3, 1805. Gauss refers to the work, which is recorded inhis mathematical diary on August 30, 1805, that later appeared in his paperSummatio Quarumdam Serierum Singularium, Comm. soc. reg. sc. Göt-ting. rec. 1(1811). Of course, he would not have written so many mathe-matical details. In the actual letter to Olbers, he was particularly delightedin a result, and surprised by its discovery, an unusual example of self-congratulation. An account of the work and this letter appears in the essayby S.J. Patterson, Gauss sums, pp. 505–528 in the collection The Shap-ing of Arithmetic after C. F. Gauss’s Disquisitiones Arithmeticae, editedby Catherine Goldstein, Norbert Schappacher, and Joachim Schwermer,Springer-Verlag, Berlin Heidelberg, 2007. The polynomials, denoted by

Gauss as .n; k/, are the q-binomial coefficients,

�n

k

�of the exercise q-

analogues.

Partitions of numbers. It was Leibniz in a letter of 1669 to Johann Bernoulliwho posed the problem of counting the number of ways a given numbercan be written as a sum of parts. Euler developed the theory of partitionsdiscovering many remarkable theorems, and establishing the approach viagenerating functions that is followed today. Among the gems that Eulerfound is his theorem relating partitions with even or odd numbers of sum-mands and the pentagonal numbers. Good places to read about these ideasare the books mentioned in the style note on q-analogues by Andrews. Ed-win’s maps of possible routes through Carré are called Ferrers diagrams,named for Norman M. Ferrers (1829–1903), an editor of the Cambridgeand Dublin Mathematical Journal, and the Quarterly Journal of Pure andApplied Mathematics. Ferrers had communicated the idea of his diagramsto J.J. Sylvester (1814–1897) who made the use of Ferrers diagrams a sig-nificant tool to understand partitions.

Take it to the limit. In this exercise I reverse history in order to relate the Gausspolynomials to the theory of partitions. By manipulating the restrictionson partitions we arrive at Euler’s product that gives the numbers p.N/ ascoefficients of the formal power series associated to the product. Euler’sfamous book Introductio analysin infinitorium, Lausanne, 1748, containsthe beginnings of partition theory (§16). Another engaging account of these

254 Style notes

ideas can be found in Lecture 3: Collecting like terms and missed oppor-tunities in Mathematical Omnibus by Dmitry Fuchs and Sergei Tabach-nikov, Amer. Math. Soc., Providence, RI, 2007. See also the book of BruceC. Berndt, Number Theory in the Spirit of Ramanujan, Amer. Math. Soc.,Providence, RI, 2006, who explores contributions of Srinivasa Ramanujan(1887–1920) to partition theory.

q-binomial theorem. The q-binomial theorem was known around 1808 by Gauss,but he did not publish it. H.A. Rothe (1773–1841) called attention to thetheorem in the preface of his book Systematisches Lehrbuch der Arithmetik,but he did not give a proof of it in the book. Among Gauss’s unpublished pa-pers, the q-binomial theorem is found in Hundert Theoreme über die neuenTranscendenten. His proof was by induction. These historical details can befound in the magisterial book of Ranjan Roy, Sources in the Developmentof Mathematics: Series and Products from the Fifteenth to the Twenty-firstCentury, Cambridge University Press, New York, 2011.

The quantum plane. This form of the binomial theorem first appeared in a paperof Marcel-Paul Schützenberger, Une interprétation de certaines solutions del’équation fonctionnelle: F.x C y/ D F.x/F.y/, C. R. Acad. Sci. Paris,236(1953), 352–353. The study of noncommuting quantities earned the ad-jective quantum in the late twentieth century. Quantum algebra became a fo-cus of much research thanks to the connection between Yang-Baxter equa-tions (physics) and knot theory. A clear introduction to this subject can befound in the book of Christian Kassel, Quantum Groups, Springer-Verlag,Heidelberg, 1995.

Scrapsheet. The identity proved here is number 3.118 in the incredible collec-tion, Combinatorial Identities, compiled by Henry W. Gould, Morgantown,W.Va., 1972. This publication is subtitled a standardized set of tables listing500 binomial coefficient summations. Reading it is like visiting a rich minewhere everywhere you turn there seems to be gold, but you have to dig itout yourself. In this exercise, I tried to follow some unlikely proof ideasuntil they fail, as if I had just transcribed a scrapsheet of work in progress.Is a mathematician someone who turns coffee into theorems? The trick ofmultiplying by 1 where 1 has a helpful representation is used. One cannotunderestimate the potential of this trick when ingeniously used. The coffeestains were designed by Freepik.com.

Mathematical Idol. Which proof do you prefer? Pitting induction against thealgebra of polynomial expressions was a lovely suggestion from my editors.

In Hana’s proof, an analogy plays the key role, namely, betweenxn � ynx � y

and.x/n � .y/nx � y . In his book Combinatorial Identities, Henry W. Gould

Style notes 255

also compares the expressions considered in the exercise to

1

x C 1

�yn

� nX

kD1

�xC1k

��yk

� :

Gould reports that the identity is known in Harry Bateman’s unpublishedNotes on Binomial Coefficients as the Capelli relation. The graphic art forthese pages was done by Anastasia Stevens.

Symmetries. The symmetric polynomials play an important role in Galois the-ory, the study of polynomials and their roots. The computations here showhow generating functions and the binomial theorem lead to expressions forxn C yn in terms of xC y and xy. This material is classical. A polynomialf .x1; x2; : : : ; xn/ is symmetric if for any permutation � of f1; 2; : : : ; ng,f .x1; x2; : : : ; xn/ D f .x�.1/; x�.2/; : : : ; x�.n//. The fundamental theoremof symmetric polynomials states that such a polynomial may be expressedas a polynomial in the polynomials s1 D x1 C x2 C � � � C xn, s2 DX

1�i<j�nxixj , s3 D

X1�i<j<k�n

xixjxk , . . . , sn D x1x2 � � � xn. For a

more classical view of how the fundamental theorem of symmetric polyno-mials fits into the study of Galois theory, I recommend reading the famoustext by George Chrystal, Algebra, in two volumes, 1st edition 1886, A. andC. Black, London; 7th edition 1999, American Mathematical Society, Prov-idence, RI. A modern account can be found in the book of Joseph Rotman,Advanced Modern Algebra, second edition, American Mathematical Soci-ety, Providence, RI, 2010.

Sums of powers. Formulas for the sums of powers were sought from antiquity.The first cases f1.nD1 C 2 C � � � C n and f2.n/ D 12 C 22 C � � � C n2

are examples of figurative numbers, in the first case triangular numbersand the second case is square pyramidal numbers. The known formulas forfk.n/ D 1k C 2k C � � � C nk were found to be polynomial in n, of de-gree k C 1. It was believed that the coefficients of such polynomials wereuniversal values to be found and explored. It was Jacob Bernoulli (1655–1705) who discovered the Bernoulli numbers and gave their first applicationto sums of powers in his posthumous 1711 book Ars Conjectandi. It is sur-prising that the Bernoulli numbers appear in so many places in mathematics.The Wikipedia page on Bernoulli numbers gives the reader some sense oftheir ubiquity. My exposition is based on the use of exponential generat-ing functions, whose product rule involves binomial coefficients. See HerbWilf’s generatingfunctionology for more details.

Parts of proof. This exercise is based on Queneau’s exercise Parties du discours(Parts of speech) in which the words of an exercise are listed according totheir part of speech, leaving the reader to put the prose back together. Aproof calls on additional components, illustrated here as the idea, method,

256 Style notes

and displays. The snake oil method of Wilf aims to establish an identitybased on sums that depend on a parameter (in this case j ). Wilf suggeststhat we construct a generating function in two variables for the function ofthe parameter given by the sum. The crucial step (the snake oil) is the in-terchange of the order of summation after which simplifications are possi-ble, leading to identities. Many illustrative examples are presented in Wilf’sbook generatingfunctionology.

Rings and ideals. The definition of a commutative ring is sufficient for a proofof the binomial theorem for nonnegative integer exponents. (Can you proveit?) Rings and their ideals are often discussed in the first undergraduatecourse on abstract algebra. If you find these ideas unfamiliar, have a lookin the book of Joseph Gallian, Contemporary Abstract Algebra, CengageLearning, Boston, MA, 9th edition, 2017. The appearance of the binomialtheorem in the proof that

pI is an ideal is an unexpected trick worth know-

ing. Ideals not only generalize the notion of a normal subgroup, but theyplay a key role in a more general notion of divisibility introduced by Kum-mer and Richard Dedekind (1831–1916) and developed by David Hilbertand Emmy Noether (1882–1935). The properties of ideals in the polyno-mial ring in several variables provide an algebraic foundation for algebraicgeometry. The radical of an ideal in such a ring plays a key role in the proofof Hilbert’s Nullstellensatz. I refer the reader to Advanced Modern Algebraby Joseph Rotman, second edition, American Mathematical Society, Provi-dence, RI, 2010, for more details.

Mathmärchen. The prose of the Grimm brothers’ fairy tales has always fasci-nated me. The tale of Herr Fix und Fertig (from the 1812/1815 collections)is the model for my tale. In the Grimm story the protagonist Mister Fix-it iskind to various animals (Frosch, frog; corbeau, crow; oshidori, duck) whohelp him win the hand of a princess in unexpected ways. The argument inthe exercise appears in Peter J. Cameron’s lovely book Combinatorics: Top-ics, Techniques, Algorithms, Cambridge University Press, New York, 1994.The more usual proof of the existence of the fields of cardinality pn usesGalois’ theory of fields and the fact that xp � x .mod p/ for all x in Z=pZ

(Fermat’s little theorem). Methods of counting in finite algebraic structurescan lead to unexpected places. The quote about generating functions andclothes lines is from Herb Wilf’s generatingfunctionology. For a thoroughintroduction to the Möbius function, see Chapter 2 of H.E. Rose, A Coursein Number Theory, Oxford Univ. Press, NY, 2nd edition, 1994.

The binomial distribution. Games of chance led to the analysis of repeatedtrials—flipping a coin or rolling a die give independent trials. Early ac-counts of the probability of k successes in n trials can be found in thework of Pascal and de Moivre. It was Jacob Bernoulli in his treatise ArsConjectandi who formalized the binomial distribution. The lure of a one-

Style notes 257

sentence exercise was irresistible, especially after reading the novel of Bo-humil Hrabal, Dancing Lessons for the Advanced in Age (1964), whichis a single sentence of 117 pages. Ed Park’s review, One sentence tells itall (New York Times, Sunday Book Review, December 24, 2010) discussesother efforts at the “Very Long Sentence.”

Wormhole points. One of the first discussions of probability theory was an ex-change of letters in 1654 between Pierre de Fermat and Blaise Pascal on theProblem of Points. In their discussion, two equally skilled players are inter-rupted while playing a game of chance for a pot of money. The score at thatmoment does not decide the winner. Based on the score and the game, howshould the pot of money be divided? A translation of the exchange betweenFermat and Pascal along with other historical background on probabilityand statistics can be found at

✇✇✇✳②♦r❦✳❛❝✳✉❦✴❞❡♣ts✴♠❛t❤s✴❤✐stst❛t✴♣❛s❝❛❧✳♣❞❢✳

According to Victor Katz, in A History of Mathematics, HarperCollins Col-lege Publishers, New York, 1993, the problem of points played a motivatingrole in Pascal’s writing of the Traité du triangle arithmétique.

You can’t always get what you want. The distribution that motivates this com-putation arises in the problem known as Banach’s matchbox problem, para-phrased in the exercise Matt Hu, graduate student. The probability in that

problem takes the form P rŒX D k� D�2n � kn � k

�1

22n�k . To know that we

have a probability distribution we need to show

nX

kD0

�2n � kn � k

�1

22n�k DnX

kD0

�nC .n � k/n � k

�1

2nCn�k

DnX

mD0

�nCm

m

� 1

2nCm

D2nX

pDn

�pn

� 12p

D 1:

The verse is based on the well-known Rolling Stones song You can’t alwaysget what you want, written by Luigi Creatore, Keith Richards, Mick Jag-ger, Hugo Peretti, and George Weiss; Copyright: Abkco Music Inc., GladysMusic, Mirage Music Int. Ltd. c/o Essex Music Int. Ltd. I learned how therefrain of this song describes mathematical research from Haynes Miller.

Matt Hu, graduate student. The problem of the distribution of the remainingnumber of candies is a reformulation of Banach’s matchbox problem that

258 Style notes

trades candies for matches (see William Feller, An Introduction to Prob-ability Theory and Its Applications, vol.1, 2nd edition, John Wiley andsons, New York, 1957, p. 157). The problem was probably introduced byHugo Steinhaus and not by Stefan Banach. Banach was a heavy smoker.Fans of Raymond Chandler will recognize the first sentence as a para-phrase of the beginning of one of his novels. Stirling’s formula, that nŠ �.2�/1=2nnC.1=2/e�n, is an extraordinary result, established in 1730. A firstapproximation was made by Abraham de Moivre around the same time.Proofs of the formula abound on the web, and center around the value oflog.nŠ/. The setting of the professor’s office recalls my advisor’s office.

Seeking successes. The probability distribution P rŒk failures, n successes� isknown as the negative binomial distribution. It also allows us to computethe waiting time to the nth success for a repeated Bernoulli trial. The distri-bution was introduced by Pascal in letters to Fermat (see Varia Opera Math-ematica. D. Petri de Fermat. Tolosae (1679)). Pierre Rémond de Montmortused the negative binomial distribution to explore the number of times a faircoin need be tossed to obtain a certain number of heads. The readers of thissort of letter were thought to be technically savvy.

TEXNO…AI�NIA. Technopaegnia is the term usually applied to classical Greekor Latin pattern poems that were written in a shape related to their content.The visual appearance of the poem added an extra dimension and additionalforce to the work. The form was revived in the Renaissance. Famous exam-ples include the poems “The Altar” and “Easter Wings” by George Her-bert (1593–1633). The form was again revived in the late 19th century ascalligrammes by Mallarmé and Apollonaire, becoming in the 20th centuryconcrete poetry to the Dadaists and later poets. Of course, a concrete poemconcerning binomial coefficients must take the shape of a triangle. Eachline has one more character than the last—punctuation doesn’t count. Eachcharacter in a mathematical expression is important. Calculus as a tool in thestudy of polynomials can never be underestimated. This proof and its gener-alizations are discussed in Combinatorial Enumeration by I.P. Goulden andD.M. Jackson, Wiley and Sons, New York, 1983 (also Dover Publications,Mineola, NY, 2004).

An area computation. This computation of area was suggested by Pascal in histreatise Potestatum numericarum summa of 1654—see Oeuvres de Pascal,edited by Brunshvicg, Boutroux, and Gazier, 1909–1914, Paris; volume 3,pp. 346–367. Another version appears in Wallis’s Arithmetica infinitorum

of 1655. The fact thatZ 1

0

xk dx D 1

k C 1was known in the years before

Newton and Leibniz introduced calculus. The role of arithmetic in resolvinggeometric problems became apparent around this time, and soon blossomsinto calculus.

Style notes 259

Experimental mathematics. This fictitious letter is written in the manner of IsaacNewton (1643–1727), who explains to “Sir” that he has discovered patternsamong the binomial coefficients that can be extended by interpolation tonegative and to fractional exponents. Furthermore, together with term-by-term integration, these series solve geometric problems in an elegant andefficient manner. The entries in the tables are from Whiteside’s The Math-ematical Papers of Isaac Newton, volume 1, Cambridge University Press,Cambridge, 1967. The table is adapted from the frontispiece of volume 1, asheet of calculations in Newton’s hand. These youthful computations re-quire rigorous proof, which Newton did not provide, though he verifiedthem by algebraic means. His investigations of calculus began with the bi-nomial series. For more about Newton and the binomial theorem, see Chap-ter VII of William Dunham in Journey through Genius, Wiley and Sons,New York, 1990; D. T. Whiteside, Newton’s discovery of the general bi-nomial theorem, Math. Gazette, 45(1961), 175–180; and Niccolò Guiccia-rdini, Isaac Newton on Mathematical Certainty and Method, MIT Press,Cambridge, MA, 2011. In the printing of the time, Newton would write ye

for “the,” yt for “that,” wch for “which,” and & for “and.”

Telescopes. Gottfried Wilhelm Leibniz (1646 –1716) is acknowledged as one ofthe discoverers of calculus with Newton. He was famously a visitor in Parisand later in London where he made an impression on the local intellectualcommunities. The summing of the reciprocals of the triangular numbers viaa telescoping sum is attributed to Leibniz, and this fictitious letter is builton his accomplishments. A discussion of Leibniz’s early mathematical ac-complishments may be found in the books Leibniz in Paris 1672–1676: HisGrowth to Mathematical Maturity by J. H. Hofmann, Cambridge Univer-sity Press, Cambridge, 1974, and The Early Mathematical Manuscripts ofLeibniz, translated from the Latin Texts published by Carl Immanuel Ger-hardt with critical and historical notes by J. M. Child, Open Court, Chicago,1920. There is a proof of the fundamental theorem of the calculus, based onthe mean value theorem, that employs a telescoping sum. The famous noteHistoria et Origo Calculi Differentialis of Leibniz, written between 1714and 1716, in which he described his contributions to the development ofcalculus, has an odd tone—Leibniz refers to himself in the third person. Myfictitious letter seems in keeping with his manner of communication. Thesignature is, in fact, an alias Leibniz used to publish a politically sensitivetreatise on the law. Historia et Origo also contains the harmonic triangle.

The Rechner. In the days before computers, lengthy and tedious calculationswere given to individuals or groups to carry out, freeing the principal inves-tigator for loftier pursuits. The term computer was coined for such folks, einRechner in German, and the term human computer is used now. D. A. Grier’sbook When Computers Were Human (Princeton University Press, 2005)

260 Style notes

tells the story of the role of human computers in determining the orbit ofHalley’s comet in 1758 and its later returns. I imagined my Rechenknechtas part of this effort. The presentation of the general binomial theorem forrational coefficients

.PCPQ/mn D P

mn Cm

nAQCm � n

2nBQCm � 2n

3nCQCm � 3n

4nDQC� � �

appears in Newton’s Epistola prior of 1676. Its effectiveness as an algo-rithm for computing roots is the point of this exercise.

Lipogram. A lipogram is a constraint on a written piece that excludes certainchosen letters of the alphabet from the piece. I have excluded the letter efrom the piece, as George Perec does in La Disparition. The story of the ex-ercise is based on the 1683 study of Jacob Bernoulli on compound interest,in which he shows that continuous compounding is bounded (Quèstionesnonnullae de usuris, cum solutione problematis de sorte alearum, propositiin Ephem. Gall. A. 1685 Acta eruditorum, 219–23). A key role is played bythe binomial coefficients and limiting expressions for them.

18th century machinations. In this exercise I have followed Vandermonde’s Mé-moire sur des irrationnelles de différents ordres avec une application aucercle, Histoire de l’Acadèmie Royale des Sciences avec les Mèmoires deMathèmatique et de Physique pour l’annèe 1772 (1772), 489–498, usingmodern notation. However, the reasoning is very 18th century in style. Sug-gestions of form lead to jumps in method, without a lot of discussion. Forexample, going from .x/n D .x/0.x � 0/n to .x/0 D .x/n.x � n/�n to theform for .y/�n, or more dramatically, the leap in 3 from .xC kC l/l .x/�lto an infinite series when k and l are rational. Without convergence re-sults 18th century mathematics moved forward based on trust in algebra.My thanks to Jim Tattersall for a copy of his article on Vandermonde anda copy of the Mémoire. Wallis’s celebrated product formula for �=2 wasmy goal of the exercise. Vandermonde also derived a formula of Newtonfor �=2 in this manner from the integrals. Term-by-term integration onlyrequires the binomial theorem as input to achieve these results.

Recipe. This exercise gives a sort of explicit recipe for the use of induction. I feelthat one ought to use induction sparingly (like cardamon or capers) becauseit usually comes after a pattern is found. Finding the pattern often leads toa proof that keeps closer to the objects at hand. This proof applies term-by-term differentiation. If a function f .x/ is represented by an infinite series,

say, f .x/ DX1

kD0ak.x�x0/k for jx�x0j < r , then at any point x1 inside

the interval of convergence .x0 � r; x0 C r/, term-by-term differentiationand integration are allowed. See the classic text of Konrad Knopp, Theoryand Application of Infinite Series, Dover Publications, Mineola, NY, 1990,

Style notes 261

for a proof. The key condition is uniform convergence of series. Differenti-ation can be useful for establishing identities between series associated withbinomials. Don’t get me wrong—induction shouldn’t be avoided! The righttool for the job makes work easier.

Math talk. Fans of National Public Radio’s Car Talk will certainly recognize thiskind of banter. The names Jack and Jake refer to Johann (1667–1748) andJacob (1655–1705) Bernoulli, brothers but also rivals in mathematics. The

operator xd

dxapplied to a power series in x has the effect of introducing a

factor of the exponent at each term:

xd

dx

1X

kD0akx

k D1X

kD0kakx

k :

And iterating the operator accumulates factors of k. This formal trick leadsto relations between generating functions and explains how powers can ariseas coefficients.

Averages and estimates. Average values and estimates play a role in some of theother exercises. They are also a tool in probabilistic methods in combina-torics. For examples, see Jiri Matoušek, and Jan Vondrák, J., The Probabilis-tic Method, lecture notes that can be found on the web, and the importantbook by Noga Alon and Joel H. Spencer, The Probabilistic Method, fourthedition, Wiley, New York, 2016. Average values and estimates of numericalfunctions also play a role in number theory. To get better estimates of thebinomial coefficients, use Stirling’s approximation for nŠ,

nŠ �p2�n

�ne

�n:

Equality. The opening question, “What does .1 C x/p2 mean?”, is something

that is not always asked in elementary calculus classes. The proposal torepresent such a function of x as a power series leads to other questions—what does it mean to represent a function by a power series? What is theradius of convergence? How does such a function depend on the exponent?I based my answers to these questions on Konrad Knopp’s exposition ofsimilar issues in the 1928 classic, Theory and Application of Infinite Series,available from Dover Publications, Minelola, NY, 1990. If we take Euler’sapproach and let .1C x/r D er ln.1Cx/, then we get a series representation

.1C x/p2 D e

p2 ln.1Cx/

D 1C ln.1C x/p2C 2 ln2.1C x/

2ŠC .

p2/3 ln3.1C x/

3ŠC � � � :

262 Style notes

Comparing this with .1 C x/r DX1

kD0

Xk

jD0s.k; j /

kŠrjxk we can ob-

tain a series representations for lnn.1 C x/. It is possible to generalize theargument in the exercise to complex exponents and a complex variable x.

� , by parts. Calculation of � has fascinated mathematicians since antiquity. Thiscalculation of � in Newtonian terms is outlined in Newton’s notes on Wal-lis’s Arithmetica Infinitorum. (See D. T. Whiteside, Newton’s discovery ofthe general binomial theorem, Math. Gazette, 45(1961), 175–180.) His bet-ter known computation uses the same sort of reasoning with the circle ofradius 1 centered at x D 1. The series of Euler was obtained around 1755via a series development of the arctan function. The beta functions appear inEuler’s work on the gamma function, which is his interpolation of the func-tion on positive integers n 7! nŠ. The integral representation of the numbersB.k; l/ offer many possible avenues of study, including integration by partsand summations. For more details about Euler’s integrals, see the Chau-venet prize winning essay of Philip J. Davis, Leonhard Euler’s Integral: ahistorical profile of the Gamma function, Amer. Math. Monthly 66(1959),849–869. Integration by parts appeared first in work of Pascal (1658) re-lating volumes and what would later become integrals. The method wasknown to Newton and Leibniz in their development of calculus.

A macaronic sonnet. Macaronic poetry refers to verse that uses several languagesat once. It is found sometimes in choral music—for example, In Dulci Ju-bilo, the Christmas carol. Is mathematics a universal language? Some be-lieve so. And consequently, no matter the language, mathematical ideas willget understood. This premise is tested in this poem. Though “unnecessary,”I provide a translation:

Definitions offer unexpected delights to us all,From the low-hanging fruit that two ideas might make fall.An example, constructed from the simplest of series,

1

1 � t D 1C t C t2 C � � � D1X

kD1tk ;

Emerges from a long look at the beta function:

B.k; l C 1/ DZ 1

0

tk�1.1 � t /l dt; defined for k � 1; l � 0.

What would happen if we multiplied the geometric series by .1 � x/l ,Then integrated over the unit interval?

1

lDZ 1

0

.1 � t /l�1 dt DZ 1

0

1

1 � t .1 � t /l dt

Style notes 263

DZ 1

0

1X

kD1tk�1

!.1 � t /l dt

D1X

kD1

Z 1

0

tk�1.1 � t /l dt D1X

kD1B.k; l C 1/:

But integration by parts allows us to immediately prove that

B.k; l C 1/ D l

kB.k C 1; l/

D � � � D l

k

l � 1k C 1

� � � l � .l � 1/k C l � 1 B.k C l; 1/

D 1

k

�k C l

l

� :

Thus we find, for n � 1, that1X

kD1

1

k

�k C l

l

� D 1

l.

A hidden integral. This identity is numbered 4.9 in Gould’s Combinatorial Iden-tities among the sums of type 1=1. The sum invites simplifications of the ex-pression, but the value was not evident to me from such maneuvers. I some-times suggest to my students that there are often only so many things wecan draw on; coaxing one of them to lead to a desired result, like 1 D k=k,is as satisfying as devising a new approach.

Letter to a princess. The first successful popularization of science by a scientistwas Leonhard Euler’s Letters to a German Princess on Subjects in Physicsand Philosophy, a collection of 234 letters written by Euler in French toFriederike Charlotte of Brandenburg-Schwedt and her sister Louise between1760 and 1762. The collected letters appeared in print between 1768 and1774 in three volumes, supported by empress Catherine II. A translationinto English was made by Henry Hunter in 1795. The subjects treated byEuler in the letters concerned physical phenomena for the most part: sound,air, temperature, light, optics, gravity, and astronomy. The letters explainedthe natural philosophy of the day. They were very popular in their time andenjoyed several editions. This exercise is based on the style of Euler’s let-ters, but it treats his mathematics, in particular his investigation of infiniteseries of the form 1� 2k C 3k � 4k C � � � . The polynomials introduced hereare known as Eulerian polynomials and their coefficients the Eulerian num-bers. More of their properties are developed in other exercises. The bookof T. Kyle Petersen, Eulerian Numbers, Birkhäuer, New York, 2015, treats

264 Style notes

the basics and some of the more subtle combinatorial appearances of thesenumbers.

Explicit Eulerian numbers. This exercise follows the example of the exercise

An explicit formula in which a formula for the binomial coefficient�n

k

�is

given only in terms of n and k. The symmetry of the binomial coefficientscan be seen in the explicit formula. The increasing property that leads tounimodality is proved in the exercise Up and down. These properties for

the Eulerian numbersDn

k

Eare established in this exercise by a comparison

of generating functions and induction on n. Worpitzky’s identity first ap-peared in the paper of J. Worpitzky Studien über die Bernoullischen undEulerischen Zahlen, J. reine angew. Math. 94(1883), 203–232.

Mutperation satticsits. Permutation statistics associate a number to each per-mutation of n letters. In the exercise we count rises, an artifact of the oneline presentation of a permutation. This number gives a filtering of the per-mutations into subsets by number of rises. This statistic appears to havebeen introduced by Percy A. MacMahon (1854–1929) in his paper Secondmemoir on the composition of numbers, Phil. Trans. Royal Soc. London,A, 207(1908), 65–134. The result was an application of his Master Theo-rem. The route from compositions of numbers—partitions of numbers intoordered pieces—to counting rises is how MacMahon discovered the con-nection between Eulerian numbers and permutations. The words have beenpermuted in this exercise, following the example of Queneau exercise, Per-mutations par groupes croissants de lettres and Permutations par groupescroissants de mots. It is surprisingly easy to unscramble the words as youread, which tells us something about how the brain wants to know what isgoing on.

Review. George Boole (1815–1864) is best known for his work on logic. How-ever, his books A Treatise on Differential Equations, MacMillan and Co.,Cambridge, 1859, and the book reviewed in the exercise became standardtextbooks in the late 19th century in England. Both are in print today asclassic texts available from Cambridge University Press. The theory of fi-nite differences continues to have a place in mathematics, especially in nu-

merical methods and dynamical systems. The difference equation��x

n

�D

�x

n � 1

�makes the binomial coefficients particularly interesting. Several of

the identities found in this book may be proved from a finite difference pointof view. Furthermore, interpolations play a role in Newton’s and Leibniz’sdiscoveries of calculus.

Plus C . This exercise follows the argument in Abel’s original paper, Beweiseines Ausdrucks, von welchem die Binomial-Formel ein einzelner Fall ist.Crelle J. 1(1826), 159–160, with the addition of modern summation nota-

Style notes 265

tion. Integration is a natural tool for induction involving polynomials be-cause it increases degree. Of course, you have to determine the constant ofintegration. Alternatively, Abel’s identity may be proved by induction andthe use of the method of undetermined coefficients. Such a proof is given inEugen Netto’s book, Lehrbuch der Combinatorik, Teubner, Leipzig, 1901.There are several generalizations of Abel’s generalization of the binomialtheorem, including one by Adolf Hurwitz, Über Abel’s Verallgemeinerungder binomischen Formel, Acta Math., 26(1902), 199–203. A survey of thesekinds of theorems may be found in the paper of A. Kelmansa, and A. Post-nikov, Generalizations of Abel’s and Hurwitz’s identities, European J. ofCombinatorics, 29(2008), 1535–1543. Googling “Abel’s binomial theorem”also obtains the form

w�1.z C w C n/n DnX

kD0

�nk

�.w C n � k/n�k�1.z C k/k :

At a carnival. Every complex number aC ib is a binomial and so when powersare taken, the binomial theorem applies. The formulas presented here aredue to Abraham de Moivre (1667–1754) around 1722, and they were provedin detail by Euler in 1749. The exercise presents something paradigmatic ofmathematical development: New viewpoints (complex numbers) simplifyformulas (trig identities) that appeared complicated. Inspiration for the styleof this exercise came from the Tom Waits’s song Step right up on his albumSmall Change.

Indicators. It is a natural question to ask for the sum of a periodic subset (forexample, every third element) of a family of binomial coefficients. I don’tbelieve the term indicator is standard, and something like character is moreusual. However, I wanted to emphasize the key property of an indicator: it isnonzero when a desired property holds, and is zero when the property doesnot hold. This exercise is based on the article by David Guichard, Sumsof selected binomial coefficients, Math. Mag., 26(1995) 209–214. There isalso an account of these ideas in Eugen Netto’s book, Lehrbuch der Com-binatorik, Teubner, Leipzig, 1901. The reader who wants to explore further

might work out the sumX

r�mjCr�n

�n

mj C r

�where 0 < r < m.

Tweets. Tweets (postings to Twitter.com) are restricted to at most 140 charac-ters (including spaces and punctuation). It is a challenge to communicate insuch an abbreviated form of discourse. This exchange is based on the wellwritten paper of Henry W. Gould, A baker’s dozen proofs of a binomialidentity, Univ. Beograd. Publ. Elek. Fak., 16(2005), 131–138. The identity

tweeted here isXn

kD0.�1/

�xk

�D .�1/n

�x � 1n

�and it appears as iden-

tity 1.5 in Gould’s book. I suggest working out the details of the proofs—by

266 Style notes

a telescoping sum (Leibniz?), induction (Fermat?), comparison of polyno-mials (Kronecker?), a special case of the Chu-Vandermonde identity, andby a complex integral (Cauchy?). Then read Gould’s paper for even moreapproaches to proving the identity. Gould’s paper illustrates nicely how in-formative different approaches to the same problem can be.

Beautiful numbers. I learned this derivation from the magisterial volume of IsaacJ. Schwatt, An Introduction to the Operations with Series, The Press of theUniversity of Pennsylvania, Philadelphia, 1924. The point of this exercise isnot really the beauty of the formula, which is certainly open to discussion.The derivation is noteworthy because so many numerical domains are vis-ited, in which new simplifications are found. The beauty of Euler’s formulalies in why it is true. The same is true for Schwatt’s identity (1.90 in Gould).

Reminiscences. This exercise is based on my memories of Emil Grosswald (1912–1989). To add one more, I had a funny, unsuccessful idea about Fermat’slast theorem that I eventually developed and wrote up. I sent a first draft ofmy ideas to Grosswald, and I was pleased to receive a letter with detailedcorrections from him. He improved the work considerably. The Egorychevmethod, sometimes called the method of coefficients, has its roots in the useof generating functions and the theory of residues. Egorychev’s book ap-peared in 1977 in Russian, appearing in 1984 in the Amer. Math. Soc. Trans-lations series. His proof of Grosswald’s identity appears on page 28. Manyother interesting identities can be found in Egorychev’s book. Another pre-sentation made be found in the paper The method of coefficieints, by D. Mer-lini, R. Sprugnoli, and M.C. Verri, Amer. Math. Monthly 114(2007), 40–57.

Winter journa.l The story Le Voyage d’hiver on which this exercise is based waswritten by Georges Perec in 1979. Since then other members of the OuLiPohave contributed an extension of Perec’s story collected in the book Win-ter Journeys by Perec and the OuLiPo, Atlas Press, London, 2013. Perec’sshort story is developed by twenty other writers. Mathematical writing en-joys this sort of potential and a publication is most prized when it spawnsmany new ideas. The mathematical relation discussed is one of the relationsof Moriarty type, as named by Harold Thayer Davis in The Summation ofSeries, Principia Press, San Antonio, 1962 (now available from Dover Pub-lications, Mineola, NY, 2015). There are also papers of Henry W. Gould onthis topic that offer other proofs of the identities of Moriarty type and moreHolmesiana. See The case of the strange binomial identities of ProfessorMoriarty, Fibonacci Quart., 10(1972), 381-391, and The design of the fourbinomial identities: Moriarty intervenes, Fibonacci Quart., 12(1974), 300-308.

Cellular automata. Cellular automata are determined by deterministic and sim-ple rules that sometimes lead to complex outcomes. The most well knownexample is the planar cellular automaton Life introduced by John Conway.

Style notes 267

(See Martin Gardner, Mathematical Games: The fantastic combinations ofJohn Conway’s new solitaire game “life,” Scientific American, 223(1970),120–123.) Using the Pascal identity (mod 2) as a generating rule leads to arelation between the Pascal triangle with the Sierpinski triangle. When pre-sented as a symmetric triangle, the Pascal triangle (mod 2) takes the formon the left and the usual portrait of the Sierpinski triangle is the array oftriangles on the right.

The study of cellular automata has been championed by Stephen Wolfram.More details about the relation between cellular automata and the binomialcoefficients may be found in the articles of Wolfram and Andrew Granvillelisted in the footnote. For more about the world of cellular automata, seeWolfram’s book A New Kind of Science, Champaign, IL, Wolfram Media,2002.

Tiling. The study of tilings of the plane has its origins in earliest times, espe-cially in the work of artists. From the tessellations of ancient Rome to theworks of the twentieth century artist M. C. Escher (1898–1972) the repe-tition of patterns can be found in most cultures. The mathematics of tilingemerged with the language of symmetry, and developed in many directionswhen aperiodic tilings were discovered. One can read deeply about tilingsin the best known book, Branko Grünbaum, G.C. Shephard, Tilings andPatterns, W.H. Freeman and Co., New York, 1987 (second edition, DoverPublications, Mineola, NY, 2016). A more recent account may be found inCharles Radin, Miles of Tiles, Student Math. Lib., vol. 1, Amer. Math. Soc.,Providence, RI, 1999. The fractal aspects of the Pascal triangle tiling are dis-cussed in chapter 8 of the book by H.-O. Peitgen, H. Jürgens, and S. Saupe,Chaos and Fractals, Springer-Verlag, New York, 1992. More advanced the-ory has been surveyed by Natalie P. Frank in A primer of substitution tilingsof the Euclidean plane, Expo. Math. 26(2008), 295–326.

Lattice points. The geometry of numbers, developed by Hermann Minkowski(1864–1909), relates the counting of lattice points to problems in numbertheory. Minkowski’s theorem provides a relation between volume of a sym-metric convex body and the number of lattice points in that body. The poly-nomial associated with dilations of a polyhedron were developed by Eu-

268 Style notes

gène Ehrhart in many papers beginning with Sur les polyhèdres rationnelshomothétiques à n dimensions, C. R. Acad. Sci. Paris 254(1962), 616–618,and his book Polynômes arithmétiques et méthode des polyhèdres en com-binatoire, Birkhäuser Verlag, Basel, 1977. Ian G. MacDonald proved thecombinatorial reciprocity of the Ehrhart polynomials in Polynomials asso-ciated with finite cell-complexes, J. London Math. Soc. 4(1971), 181–192.A clear and engaging account of these ideas may be found in the bookof Matthias Beck and Sinai Robins, Computing the Continuous Discretely,Integer-point Enumeration in Polyhedra, UTM, Springer-Verlag, New York,second edition, 2015.

Afterlife. Both Newton and Abel studied the binomial series deeply. The ques-tion of convergence emerged as the methods of calculus were developed inthe 18th century. Abel’s 1826 paper, Untersuchungen über die Reihe:

1C m

1x C m � .m � 1/

1 � 2 � x2 C m � .m � 1/ � .m � 2/1 � 2 � 3 � x3 C � � � usw.,

Crelle J. 1(1826), 311–339, gave the first rigorous proof of convergence(when jxj < 1) and, though flawed in the assumption of uniform conver-gence, indicated what is needed in order to be certain of the use of infiniteseries when representing a function. Some of the lines in this dialogue aredirect quotes or slight paraphrases from the writings of Newton and Abel.See the book of Richard Westfall, Never at Rest : a Biography of IsaacNewton, Cambridge University Press, New York, 1980, and the work ofHenrik Kragh Sørensen, especially the paper Exceptions and counterexam-ples: understanding Abel’s comment on Cauchy’s Theorem, Historia Math-ematica 32(2005), 453–480. Many of the quotes from Abel are found inletters to his teacher and friend B. M. Holmboe (1795–1850). A change instyle from mathematics as natural philosophy to pure mathematics certainlytakes place in the period between the lives of the interlocutors, and consid-ering their contributions to the history of the binomial theorem points upthis change. The afterlife setting of the dialogue is inspired by the wonder-ful book by David Eagleman, Sum: Forty Tales from the Afterlives (Vintage,New York, 2010).

Matt Hu and the Euler caper. This exercise was inspired by the lovely articleof James Tanton, A dozen questions about Pascal’s triangle, Math Hori-zons, 16(2008), 5–30. My proof uses the Euler characteristic as a tool todo the counting. Tanton has another proof of the final formula. The pa-per of M. Noy, A short solution of a problem in combinatorial geometry,Math Mag. 69(1996), 52–53, also establishes the formula from the Eulercharacteristic. The Euler characteristic plays an important role in combina-torics and in topology. For an introduction to many applications of the Eulercharacteristic, see the excellent book Euler’s Gem: The Polyhedron Formula

Style notes 269

and the Birth of Topology by David Richeson, Princeton University Press,Princeton, NJ, 2008.

Hypergeometric musings. Euler introduced the hypergeometric series 2F1 as asolution to a second order differential equation. He developed transforma-tion rules among such series in his paper Specimen transfromationis sin-gularis serierum, Nova Acta Acad. Petropol. 7(1778) 58–70, also in OperaOmnia, series 2, 16, 41–55 (E710). Gauss extended these researches follow-ing some developments by his PhD advisor, Johann Friedrich Pfaff (1765–

1825). His paper Disquisitiones Generales Circa Seriem Infinitam�˛ˇ

.1 �

�x

C�˛.˛ C 1/ˇ.ˇ C 1/

.1 � . C 1//

�x2C

�˛.˛ C 1/.˛ C 2/ˇ.ˇ C 1/.ˇ C 2/

.1 � . C 1/. C 2/

�x3C etc. Pars

Prior, Comm. Soc. Regiae Sci. Gottingensis Recentiores, Vol. II. 1812(reprinted in Gesammelte Werke, Bd. 3, 123–163 and 207–229, 1866) con-tains his systematic development of hypergeometric series. The case ofhigher degree polynomials P.k/ and Q.k/, or generalized hypergeomet-ric functions, was introduced by Leo August Pochhammer in Hyperge-ometrische Functionen nter Ordnung, J. reine angew. Math. 71(1870) 316–352. Pochhammer also introduced notation for the hypergeometric func-tions. The relations between binomial identities and hypergeometric se-ries has long been recognized. A nice introduction is the paper of RanjanRoy, Binomial identities and hypergeometric series, Amer. Math. Monthly,94(1987), 36–46. This connection may be made more formal, and it sup-plies an ingredient in the work of Wilf and Zeilberger on the verifiability ofbinomial identities.

On the bus. The story told here will be familiar to anyone who has opened Ray-mond Queneau’s Execices de Style. The inspiration for the t-shirt is a photoon Doron Zeilberger’s website, showing him in just such a t-shirt togetherwith the question “Who you gonna call?” The Pfaff-Saalschütz Theoremwas known around 1797 by Pfaff and it was rediscovered by L. Saalschützin 1890. A nice proof may be found in the paper of J. Dougall, On Van-dermonde’s theorem and some more general expansions, Proc. EdinburghMath. Soc. 25(1907), 33–47. Passing the identity through a test for beinga hypergeometric series is a simple case of the method of Sister CelineFasenmeyer (1906–1996) introduced in her PhD thesis Some generalizedhypergeometric polynomials, Bull. Amer. Math. Soc. 53 (1947), 806–812(1945 University of Michigan PhD thesis). The method was extended byR.W. Gosper, Decision procedure for indefinite hypergeometric summation,Proc. Natl. Acad. Sci. USA 75(1978), 40–42, and by Herb Wilf and DoronZeilberger, An Algorithmic Proof Theory for Hypergeometric (Ordinaryand “q”) Multisum/Integral Identities. Invent. Math. 108(1992), 575–633.Their method is far-reaching and can be used to discover identities algorith-mically. A description of these developments can be found in the book of

270 Style notes

M. Petkovsek, H.S. Wilf, D. Zeilberger, A=B, A K Peters (1996). Shaloshmentioned in the exercise is Shalosh B. Ekhad, a distinguished colleague ofDoron Zeilberger.

I came to depend on a few books that have not been mentioned because theytreat the foundations of combinatorial thinking. I cannot fail to mention

Louis Comtet, Advanced Combinatorics: The Art of Finite and Infinite Ex-pressions, D. Reidel Pub. Co. Boston, 1974, a translation by J.W. Neinhuys ofAnalyse Combinatoire, tomes I et II, Presses Universitaires de France, Paris,1970;

John Riordan, Introduction to Combinatorial Analysis, Dover Publications,Mineola, NY, 2002, an unabridged republication of the work published by JohnWiley & Sons, Inc., New York, 1958;

Ron L. Graham, Donald E. Knuth, and O. Patashnik, Concrete Mathemat-ics, Addison-Wesley Pub. Co., Reading, MA, 1990.

I recommend these books for further exploration of combinatorial ideas. Thequote from Ian Hacking in the preface is from his paper ‘Style’ for Historiansand Philosophers. Studies in the History and Philosophy of Science, 23(1992),1–20. The Seven-Percent Solution by Nicholas Meyer was published in 1974 byE.P. Dutton, New York. The mathematical papers of Raymond Queneau are Surles suites s-additives, C. R. Acad. Sci. Paris Sér. A-B 266(1968) A957–A958,and Sur les suites s-additives, J. Combinatorial Theory Ser. A 12(1972), 31–71.Finally, for English readers, there is a superb translation (with some additions)of Queneau’s Exercices de Style, published as Exercises in Style, and translatedby Barbara Wright and Chris Clarke, New Directions, New York, NY, 2013.

Index

�n

k

�, 1

Tn, 11Pn, 11ŒN�, 39�X

k

�, 41

L<nCk�1;k , 41�Œn�

2

�, 42

L�n;k

, 42

R.k; l/, 44Dn, 51MN , 56� n

a1; a2; : : : ; at

�, 57

�n

k

�m

, 59, 90

a � b .mod p/, 63ep.n/, 65brc, 65, 206Yn

kD0, 73

Hn, 741 � 2 � � �bi � � � .n � 1/ � n, 77Y

iai , 83�

x

k

�, 86

�p=q

k

�, 88

.x/n, 93, 131, 170s.n; k/, 94, 181S.n; k/, 94, 101

P D��

n

k

��, 96

S , 97M 0, 97SUR.Œn�; Œk�/, 100HOM.Œn�; Œk�/, 100

, 100Cn, 104, 107Œn�Šq , 109�n

k

�, 109, 116, 118, 123

.n; k/, 112pn;k.N /, 116, 118pk.N /, 118p.N/, 119FN .q; t/, 120Bk , 136R=I , 140pI , 141

Fp , 142�.n/, 144E ŒX�, 145, 153

Pmn , 166

Œpn� , 170

xd

dx, 177

� , 184, 185B.k; l/, 184, 186Dn

k

E, 192, 192

†n, 195�f .x/, 197Ef .x/, 198X

f .x/, 199

!m, 206z, 207b�1, 213�n, 224L.P;N /, 226V �E C F , 231

pFq

�a1; a2; : : : ; apb1; b2; : : : ; bq

I z�

, 234

271

272 Index

Abel, Niels Henrik (1802–1829), 200, 227,250, 264, 268

absorption identity, 8, 20, 59, 111, 146actuaries, 155alchemy, 74, 248algebra, 5, 16, 27, 57, 86, 90, 126, 140,

142, 170, 195algorithm, 79, 81alternating sum of powers, 189approximation, 166, 179area under a curve, 158arithmetical triangle, 1, 6, 24, 161averages, 178

Babbage, Charles (1791–1871), 79Banach, Stefan (1892–1945), 258Banach’s matchbox problem, 152, 258Bernoulli numbers, 136, 248, 255Bernoulli, Jacob (1655–1705), 255, 256,

260Bernoulli, Johann (1667–1748), 253, 261Bertrand’s Postulate, 83, 249beta function, 184, 186, 262bijective proof, 18, 26, 41, 241, 244binary counting, 23, 26, 242binomial coefficients modulo a prime, 71,

247binomial distribution, 145, 256binomial inversion, 49, 52, 96, 100, 244,

251Blom, L. 215Boole, George (1815–1864), 197, 264bounds, 179Bourbaki, 99, 251

calculus, 75, 157, 174, 176, 183, 197, 259,268

Carré, 9, 25, 28, 115, 239, 242carry, 67Catalan numbers, 106, 108, 251Catalan, Eugene C. (1814–1894), 108, 252Cauchy, Auguste (1789–1857), 180, 213,

250, 266cellular automata, 218, 266central binomial coefficient, 77, 83, 178chair-committee identity, 7change-of-basis, 94

Chebyshev polynomials, 205, 216choices with repetition, 37, 42, 102, 143Chu Shih-Chieh (13th century), 24, 27, 242Chu-Vandermonde identity, 27, 87, 209closed form, 20, 61, 66, 106, 126, 157, 184,

192, 210coffee, 126combinatorial reciprocity, 226, 268commutative ring, 140complete graph, 43complex integrals, 209, 213, 266complex numbers, 204, 206, 210, 213compound interest, 168concrete poetry, 157, 258conjugation, 207, 210convergence, 88, 174, 180, 228, 261, 268counting, 3, 7, 9, 18, 22, 25, 28, 31, 37,

39, 46, 51, 54, 59, 100, 107, 115, 118,147, 195, 224, 232

de Moivre, Abraham (1667–1754), 90, 216decision tree, 4derangement number, 51derangement, 51, 245derivative, 157, 174, 176, 190dialogue, 104, 175, 227Dies Irae, 14, 240distribution problem, 31, 59divergent series, 189, 228divisibility, 62, 65, 75, 77, 81Doyle, Arthur Conan (1859–1930), 215Egorychev’s method, 214, 266Ehrhart polynomials, 225, 268Erdos, Pàl (1913–1996), 68, 83, 247, 249

estimates, 29, 83, 154, 179, 261Euclid, 12, 240Euler characteristic, 231, 268Euler integral of the first kind, 184, 187Euler, Leonhard (1707–1783), 1, 108, 119,

184, 189, 204, 234, 246, 250, 252,253, 262, 263, 265, 268

Euler’s formula, 210Eulerian numbers, 192, 192, 196, 264expected value, 145, 153, 156explicit formula, 20, 22, 52, 106, 135, 192

Index 273

exponential generating function, 135, 138,255

fairy tale, 142falling factorial, 93, 170, 198, 242Fasenmeyer, Celine (1906–1996), 269Fermat, Pierre de (1601–1665), 63, 246,

257Fermat’s little theorem, 64, 71, 142, 246,

256Ferrer diagrams, 115, 253Ferrers, Norman M. (1829–1903), 253fibomomials, 243Fibonacci sequence, 35, 243figurative numbers, 11, 24, 240, 255finite differences, 167, 197, 264finite field, 71, 142finite integral, 199Flatland, 240Fleming, Ian (1908–1962), 215formula of Legendre, 66, 84formula of Touchard, 56, 245formulas for � , 162, 172, 184, 262fundamental theorem, symmetric polyno-

mials, 134

games, 43, 244Gauss identity, 234Gauss sum, 112Gauss, Carl Friedrich (1777–1855), 63, 71,

109, 112, 234, 246, 253, 254, 269Gaussian polynomials, 109, 112, 118, 123,

253, 254generalized binomial theorem, 88, 161,

180, 249generalized hypergeometric functions,

233, 236, 269generating function, 102, 104, 116, 120,

132, 135, 138, 143, 190, 251geometric representation, 12geometric series, 88, 132, 173, 186, 190,

216Glaisher’s theorem, 72graduate school, 152Granville, Andrew (1962–), 69, 82, 220,

247, 248, 249, 267graphic novel, 129

Grosswald, Emil (1912–1989), 212, 266

haiku, 62, 246Halley, Edmond (1656–1742), 166Hardy, G.H. (1877–1947), 247, 249harmonic numbers, 74, 77harmonic triangle, 164, 259hidden variable, 176higher parabolas, 158hockey stick relation, 34, 39, 111Holmes, Sherlock, vii, 215human computers, 166, 259Huygens, Christiaan (1629–1695), 163,

215hypergeometric series, 233, 236, 269

ideal, 140, 256indicator, 206, 265induction, viii, 33, 35, 38, 45, 63, 83, 86,

121, 124, 129, 150, 159, 168, 173,190, 193, 201, 209, 222, 231, 246,252, 260

inequalities, 178integer lattice, 224integration by parts, 183, 186interpolation, 161, 167, 197, 259, 266inversion, seebinomialinversionirrational exponents, 180irreducible polynomials, 142

joke, 273

Kemeny, John G. (1926–1992), 97, 251Kummer, Ernst (1810–1893), 67, 256Kummer’s theorem, 67, 68, 222, 247

lattice paths, 9, 25, 28Laurent series, 213LeBlanc, M. (Germain, Sophie) (1776–

1831), 112Legendre, Adrien-Marie (1752–1833), 66Leibniz, Gottfried (1646–1716), 163, 217,

245, 253, 259Leonardo of Pisa (ca. 1170– ca. 1250), 35,

243letter, 51, 112, 155, 161, 163, 189limerick, 29, 242

274 Index

limits, 52, 118, 169, 180, 189linear algebra, 93, 96, 250lipogram, viii, 168, 260long division, 131Lucas’s theorem, 72, 247

macaronic poem, 186, 262MacMahon, Percy A. (1854–1929), 264Mann–Shanks theorem, 81, 249matrix derivative, 97Matt Hu, 152, 209, 230, 257ménage numbers, 56, 245method of coefficients, 214, 266milk, 209Minkowski, Herrmann (1864–1909), 267modular arithmetic, 63, 71, 79, 247Möbius inversion/function, 49, 144, 256Moriarty identity, 216, 235, 266Moriarty, James, vii, 215multinomial coefficients, 57, 60, 91, 245multiplicative principle, 3, 7, 18, 22, 239,

241music, 14

n-simplex, 224negative binomial distribution, 155, 257Newton, Isaac (1643–1727), 15, 89, 161,

166, 183, 227, 240, 259, 260, 262,268

Newton’s binomial theorem, 105, 166, 180,250

Noether, Emmy (1882–1935), 256notation, 1, 33, 57, 63, 71, 73, 97, 116, 234,

239number of poker hands, 22

Olbers, Heinrich W. (1758–1840), 112,253

one sentence, 145, 257Oresme, Nicole (1320–1382), 74, 248

p-adic representation, 65, 69, 72, 222, 247partition of a set, 99partitions of numbers, 116, 118, 120, 253,

253, 264parts of speech, 138, 255

Pascal identity, 5, 7, 10, 15, 33, 37, 55, 86,97, 218, 244

—for multinomials, 58—for other numbers, 59, 90, 109, 113, 124,

192Pascal triangle, 1, 17, 21, 23, 30, 59, 68,

161, 223, 240, 250Pascal-de Moivre triangle, 59, 90, 250Pascal, Blaise (1623–1662), vii, 24, 159,

239, 257, 258Perec, Georges (1936–1982)), viii, 260,

266permutations, 51, 195, 255, 264Pfaff-Saalschütz identity, 237, 269Pingala (ca. 200 B.C.E.), 23, 242Pochhammer, Leo A. (1841–1920), 269Pochhammer symbol, 93, 100, 109, 131,

170, 242, 254polynomial identities, 86, 94, 103, 109,

112, 126, 129, 132, 170, 190, 200polynomial rings, 142, 256polynomials in two variables, 132polytope, 224powers of 11, 15prime numbers, 77, 79, 81, 248, 249principle of inclusion and exclusion (PIE),

46, 55, 60, 244probability distribution, 145, 150, 153,

155, 256, 257problème des ménages, 54, 245problem of points, 147, 257proof by contradiction, 68, 247proof without words, 11, 240pseudocode, 79

q-binomial coefficients, 109, 116, 118,120, 252, 254

q-Chu-Vandermonde identity, 122, 125quadratic formula, 105, 252quantum binomial theorem, 123, 254Queneau, Raymond (1903–1976), vii, 147,

244, 255, 269

radical of an ideal, 141Ramanujan, Srinivasa (1887–1920), 254Ramsey theory, 43, 244rearrangement theorem, 181

Index 275

recipe, 173, 260reciprocals of binomial coefficients, 163,

185, 186, 187recurrence relation, 33, 35, 37, 104, 108,

117, 137, 192, 195, 243reductio ad absurdum, 68, 247repetitions, 37, 42, 102, 143, 244, 251restricted distribution, 59, 116, 246restricted sums, 206ring, 140rises, 195rising factorial, 233Rolling Stones, 257roots of unity, 113, 206row sums, 16, 18, 26rule of sum, 7, 239

Sanskrit prosody, 23, 241Shalosh B. Ekhad, 237shift matrix, 97shifted arithmetic triangle, 81Sierpinski triangle, 218, 267slack variables, 225snake oil method, 138, 255sonnet, 186squarefree numbers, 68, 247Stirling numbers, 94, 101, 181, 250, 251Stirling’s formula, 154, 258, 261subset-of-a-subset identity, 19, 49, 127,

241substitution tilings, 221summation notation, 5

sums of powers, 135, 160, 189, 255surjection, 99Sylvester, James Joseph (1814–1897), 253symmetric expressions, 16, 18, 29, 132,

193, 255

tanka, 62, 246technopaegnia, 157, 258telescoping sum, 69, 159, 163, 209, 259tetrahedral numbers, 11theorem of Fine, 73theory of residues, 213, 266tiling the plane, 221, 267topology, 231, 268trees, 4, 107, 252triangular numbers, 11, 135, 159, 163, 240,

255, 259trigonometry, 114, 203, 207, 211Twitter, 209, 265

unimodal, 29, 193, 243, 264

Vandermonde, Alexandre-T. (1735–1796),27, 170, 242, 260

von Ettinghausen, Andreas (1796–1878), 1

Wallis, John (1616–1703), 162, 172, 258,260

Wilson’s theorem, 77, 248Wolstenholme, Joseph (1829–1891), 77,

248Wolstenholme’s theorem, 77

What

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