partition regularity
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Partition regularityFrom Wikipedia, the free encyclopedia
Chapter 1
Multiplicative partition
In number theory, a multiplicative partition or unordered factorization of an integer n that is greater than 1 isa way of writing n as a product of integers greater than 1, treating two products as equivalent if they differ only inthe ordering of the factors. The number n is itself considered one of these products. Multiplicative partitions closelyparallel the study of multipartite partitions, discussed in Andrews (1976), which are additive partitions of finitesequences of positive integers, with the addition made pointwise. Although the study of multiplicative partitions hasbeen ongoing since at least 1923, the name “multiplicative partition” appears to have been introduced by Hughes& Shallit (1983). The Latin name “factorisatio numerorum” had been used previously. MathWorld uses the termunordered factorization.
1.1 Examples
• The number 20 has four multiplicative partitions: 2 × 2 × 5, 2 × 10, 4 × 5, and 20.
• 3 × 3 × 3 × 3, 3 × 3 × 9, 3 × 27, 9 × 9, and 81 are the five multiplicative partitions of 81 = 34. Because it isthe fourth power of a prime, 81 has the same number (five) of multiplicative partitions as 4 does of additivepartitions.
• The number 30 has five multiplicative partitions: 2 × 3 × 5 = 2 × 15 = 6 × 5 = 3 × 10 = 30.
• In general, the number of multiplicative partitions of a squarefree number with i prime factors is the ith Bellnumber, Bᵢ.
1.2 Application
Hughes& Shallit (1983) describe an application ofmultiplicative partitions in classifying integers with a given numberof divisors. For example, the integers with exactly 12 divisors take the forms p11, p×q5, p2×q3, and p×q×r2, where p,q, and r are distinct prime numbers; these forms correspond to the multiplicative partitions 12, 2×6, 3×4, and 2×2×3respectively. More generally, for each multiplicative partition
k =∏
ti
of the integer k, there corresponds a class of integers having exactly k divisors, of the form
∏pti−1i ,
where each pi is a distinct prime. This correspondence follows from themultiplicative property of the divisor function.
2
1.3. BOUNDS ON THE NUMBER OF PARTITIONS 3
1.3 Bounds on the number of partitions
Oppenheim (1926) credits McMahon (1923) with the problem of counting the number of multiplicative partitionsof n; this problem has since been studied by other others under the Latin name of factorisatio numerorum. If thenumber of multiplicative partitions of n is an, McMahon and Oppenheim observed that its Dirichlet series generatingfunction ƒ(s) has the product representation
f(s) =∞∑
n=1
anns
=∞∏k=2
1
1− k−s.
The sequence of numbers an begins
1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, ... (sequence A001055 in OEIS).
Oppenheim also claimed an upper bound on an, of the form
an ≤ n
(exp logn log log lognlog logn
)−2+o(1)
,
but as Canfield, Erdős & Pomerance (1983) showed, this bound is erroneous and the true bound is
an ≤ n
(exp logn log log lognlog logn
)−1+o(1)
.
Both of these bounds are not far from linear in n: they are of the form n1−o(1). However, the typical value of an ismuch smaller: the average value of an, averaged over an interval x ≤ n ≤ x+N, is
a = exp(
4√logN√
2e log logN(1 + o(1)
)),
a bound that is of the form no(1) (Luca, Mukhopadhyay & Srinivas 2008).
1.4 Additional results
Canfield, Erdős & Pomerance (1983) observe, and Luca, Mukhopadhyay & Srinivas (2008) prove, that most numberscannot arise as the number an of multiplicative partitions of some n: the number of values less than N which arise inthis way is NO(log log log N / log log N). Additionally, Luca, Mukhopadhyay & Srinivas (2008) show that most values of nare not multiples of an: the number of values n ≤ N such that an divides n is O(N / log1 + o(1) N).
1.5 See also• partition (number theory)
• divisor
1.6 References• Andrews, G. (1976), The Theory of Partitions, Addison-Wesley, chapter 12.
• Canfield, E. R.; Erdős, Paul; Pomerance, Carl (1983), “On a problem of Oppenheim concerning “factorisationumerorum"", Journal of Number Theory 17 (1): 1–28, doi:10.1016/0022-314X(83)90002-1.
4 CHAPTER 1. MULTIPLICATIVE PARTITION
• Hughes, John F.; Shallit, Jeffrey (1983), “On the number of multiplicative partitions”, American MathematicalMonthly 90 (7): 468–471, doi:10.2307/2975729, JSTOR 2975729.
• Knopfmacher, A.; Mays, M. (2006), “Ordered andUnordered Factorizations of Integers”,Mathematica Journal10: 72–89. As cited by MathWorld.
• Luca, Florian; Mukhopadhyay, Anirban; Srinivas, Kotyada (2008), On the Oppenheim’s “factorisatio numero-rum” function, arXiv:0807.0986.
• MacMahon, P. A. (1923), “Dirichlet series and the theory of partitions”, Proceedings of the London Mathe-matical Society 22: 404–411, doi:10.1112/plms/s2-22.1.404.
• Oppenheim, A. (1926), “On an arithmetic function”, Journal of the London Mathematical Society 1 (4): 205–211, doi:10.1112/jlms/s1-1.4.205.
1.7 Further reading• Knopfmacher, A.; Mays, M. E. (2005), “A survey of factorization counting functions” (PDF), InternationalJournal of Number Theory 1 (4): 563–581, doi:10.1142/S1793042105000315
1.8 External links• Weisstein, Eric W., “Unordered Factorization”, MathWorld.
Chapter 2
Partition (number theory)
Young diagrams associated to the partitions of the positive integers 1 through 8. They are arranged so that images under the reflectionabout the main diagonal of the square are conjugate partitions.
In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way ofwriting n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the
5
6 CHAPTER 2. PARTITION (NUMBER THEORY)
Partitions of n with biggest addend k
same partition. (If order matters, the sum becomes a composition.) For example, 4 can be partitioned in five distinctways:
43 + 12 + 22 + 1 + 11 + 1 + 1 + 1
The order-dependent composition 1 + 3 is the same partition as 3 + 1, while the two distinct compositions 1 + 2 + 1and 1 + 1 + 2 represent the same partition 2 + 1 + 1.A summand in a partition is also called a part. The number of partitions of n is given by the partition function p(n).So p(4) = 5. The notation λ ⊢ n means that λ is a partition of n.Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number ofbranches of mathematics and physics, including the study of symmetric polynomials, the symmetric group and ingroup representation theory in general.
2.1 Examples
The seven partitions of 5 are:
• 5
2.2. REPRESENTATIONS OF PARTITIONS 7
• 4 + 1
• 3 + 2
• 3 + 1 + 1
• 2 + 2 + 1
• 2 + 1 + 1 + 1
• 1 + 1 + 1 + 1 + 1
In some sources partitions are treated as the sequence of summands, rather than as an expression with plus signs. Forexample, the partition 2 + 2 + 1 might instead be written as the tuple (2, 2, 1) or in the even more compact form (22,1) where the superscript indicates the number of repetitions of a term.
2.2 Representations of partitions
There are two common diagrammatic methods to represent partitions: as Ferrers diagrams, named after NormanMacleod Ferrers, and as Young diagrams, named after the British mathematician Alfred Young. Both have severalpossible conventions; here, we use English notation, with diagrams aligned in the upper-left corner.
2.2.1 Ferrers diagram
The partition 6 + 4 + 3 + 1 of the positive number 14 can be represented by the following diagram:The 14 circles are lined up in 4 rows, each having the size of a part of the partition. The diagrams for the 5 partitionsof the number 4 are listed below:
2.2.2 Young diagram
Main article: Young diagram
An alternative visual representation of an integer partition is its Young diagram. Rather than representing a partitionwith dots, as in the Ferrers diagram, the Young diagram uses boxes or squares. Thus, the Young diagram for thepartition 5 + 4 + 1 iswhile the Ferrers diagram for the same partition is
While this seemingly trivial variation doesn't appear worthy of separate mention, Young diagrams turn out to beextremely useful in the study of symmetric functions and group representation theory: in particular, filling the boxesof Young diagrams with numbers (or sometimes more complicated objects) obeying various rules leads to a familyof objects called Young tableaux, and these tableaux have combinatorial and representation-theoretic significance.[1]As a type of shape made by adjacent squares joined together, Young diagrams are a special kind of polyomino.[2]
2.3 Partition function
In number theory, the partition function p(n) represents the number of possible partitions of a natural number n,which is to say the number of distinct ways of representing n as a sum of natural numbers (with order irrelevant). Byconvention p(0) = 1, p(n) = 0 for n negative.The first few values of the partition function are (starting with p(0)=1):
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255,1575, 1958, 2436, 3010, 3718, 4565, 5604, … (sequence A000041 in OEIS).
8 CHAPTER 2. PARTITION (NUMBER THEORY)
The exact value of p(n) has been computed for large values of n, for example p(100) = 190,569,292, p(1000) is24,061,467,864,032,622,473,692,149,727,991 or approximately 2.40615×1031,[3] and p(10000) is 36,167,251,325,...,906,916,435,144or approximately 3.61673×10106.As of June 2013, the largest known prime number that counts a number of partitions is p(120052058), with 12198decimal digits.[4]
2.3.1 Generating function
The generating function for p(n) is given by:[5]
∞∑n=0
p(n)xn =∞∏k=1
(1
1− xk
).
Expanding each factor on the right-hand side as a geometric series, we can rewrite it as
(1 + x + x2 + x3 + ...)(1 + x2 + x4 + x6 + ...)(1 + x3 + x6 + x9 + ...) ....
The xn term in this product counts the number of ways to write
n = a1 + 2a2 + 3a3 + ... = (1 + 1 + ... + 1) + (2 + 2 + ... + 2) + (3 + 3 + ... + 3) + ...,
where each number i appears ai times. This is precisely the definition of a partition of n, so our product is the desiredgenerating function. More generally, the generating function for the partitions of n into numbers from a set A can befound by taking only those terms in the product where k is an element of A. This result is due to Euler.The formulation of Euler’s generating function is a special case of a q-Pochhammer symbol and is similar to theproduct formulation of many modular forms, and specifically the Dedekind eta function.The denominator of the product is Euler’s function and can be written, by the pentagonal number theorem, as
(1− x)(1− x2)(1− x3) · · · = 1− x− x2 + x5 + x7 − x12 − x15 + x22 + x26 − . . . .
2.3. PARTITION FUNCTION 9
where the exponents of x on the right hand side are the generalized pentagonal numbers; i.e., numbers of the form½m(3m − 1), where m is an integer. The signs in the summation alternate as (−1)|m|−1 . This theorem can be usedto derive a recurrence for the partition function:
p(k) = p(k − 1) + p(k − 2) − p(k − 5) − p(k − 7) + p(k − 12) + p(k − 15) − p(k − 22) − ...
where p(0) is taken to equal 1, and p(k) is taken to be zero for negative k.
2.3.2 Congruences
Main article: Ramanujan’s congruences
Srinivasa Ramanujan is credited with discovering that congruences in the number of partitions exist for argumentsthat are integers ending in 4 and 9.[6]
p(5k + 4) ≡ 0 (mod 5)
For instance, the number of partitions for the integer 4 is 5. For the integer 9, the number of partitions is 30; for 14there are 135 partitions. This is implied by an identity, also by Ramanujan,[7][8]
∞∑k=0
p(5k + 4)xk = 5(x5)5∞(x)6∞
where the series (x)∞ is defined as
(x)∞ =∞∏
m=1
(1− xm).
He also discovered congruences related to 7 and 11:[9]
p(7k + 5) ≡ 0 (mod 7)
p(11k + 6) ≡ 0 (mod 11).
and for p=7 proved the identity[8]
∞∑k=0
p(7k + 5)xk = 7(x7)3∞(x)4∞
+ 49x(x7)7∞(x)8∞
Since 5, 7, and 11 are consecutive primes, one might think that there would be such a congruence for the next prime13, p(13k+ a) ≡ 0 (mod 13) for some a. This is, however, false. It can also be shown that there is no congruence of theform p(bk+ a) ≡ 0 (mod b) for any prime b other than 5, 7, or 11.In the 1960s, A. O. L. Atkin of the University of Illinois at Chicago discovered additional congruences for smallprime moduli. For example:
p(113 · 13 · k + 237) ≡ 0 (mod 13).
Ono (2000) proved that there are such congruences for every prime modulus. Later, Ahlgren & Ono (2001) showedthere are partition congruences modulo every integer coprime to 6.
10 CHAPTER 2. PARTITION (NUMBER THEORY)
2.3.3 Partition function formulas
Recurrence formula
Main article: Pentagonal number theorem
Leonhard Euler's pentagonal number theorem implies the identity
p(n) = p(n− 1) + p(n− 2)− p(n− 5)− p(n− 7) + · · ·
where the numbers 1, 2, 5, 7, ... that appear on the right side of the equation are the generalized pentagonal numbersgk = k(3k−1)
2 for nonzero integers k. More formally,
p(n) =∑k =0
(−1)k−1p (n− k(3k − 1)/2)
where the summation is over all nonzero integers k (positive and negative) and p(m) is taken to be 0 if m < 0.
Approximation formulas
Approximation formulas exist that are faster to calculate than the exact formula given above.An asymptotic expression for p(n) is given by
p(n) ∼ 1
4n√3exp
(π
√2n
3
)as n → ∞.
This asymptotic formula was first obtained by G. H. Hardy and Ramanujan in 1918 and independently by J. V.Uspensky in 1920. Considering p(1000), the asymptotic formula gives about 2.4402 × 1031, reasonably close to theexact answer given above (1.415% larger than the true value).Hardy and Ramanujan obtained an asymptotic expansion with this approximation as the first term:
p(n) =1
2π√2
v∑k=1
√k Ak(n)
d
dnexp
(π
k
√2
3
(n− 1
24
)),
where
Ak(n) =∑
0≤m<k; (m, k)= 1
eπi[s(m, k) − 1k 2nm].
Here, the notation (m, n) = 1 implies that the sum should occur only over the values of m that are relatively prime ton. The function s(m, k) is a Dedekind sum.The error after v terms is of the order of the next term, and vmay be taken to be of the order of√n . As an example,Hardy and Ramanujan showed that p(200) is the nearest integer to the sum of the first v=5 terms of the series.In 1937, Hans Rademacher was able to improve on Hardy and Ramanujan’s results by providing a convergent seriesexpression for p(n). It is[10]
p(n) =1
π√2
∞∑k=1
√k Ak(n)
d
dn
1√n− 1
24
sinh[π
k
√2
3
(n− 1
24
)] .
2.4. RESTRICTED PARTITIONS 11
The proof of Rademacher’s formula involves Ford circles, Farey sequences, modular symmetry and the Dedekind etafunction in a central way.It may be shown that the k-th term of Rademacher’s series is of the order
exp(π
k
√2n
3
),
so that the first term gives the Hardy–Ramanujan asymptotic approximation.Paul Erdős published an elementary proof of the asymptotic formula for p(n) in 1942.[11][12]
Techniques for implementing the Hardy-Ramanujan-Rademacher formula efficiently on a computer are discussed inJohansson,[13] where it is shown that p(n) can be computed in softly optimal time O(n1/2+ε). The largest value of thepartition function computed exactly is p(1020), which has slightly more than 11 billion digits.[14]
2.4 Restricted partitions
In both combinatorics and number theory, families of partitions subject to various restrictions are often studied.[15]This section surveys a few such restrictions.
2.4.1 Conjugate and self-conjugate partitions
If we now flip the diagram of the partition 6 + 4 + 3 + 1 along its main diagonal, we obtain another partition of 14:By turning the rows into columns, we obtain the partition 4 + 3 + 3 + 2 + 1 + 1 of the number 14. Such partitions aresaid to be conjugate of one another.[16] In the case of the number 4, partitions 4 and 1 + 1 + 1 + 1 are conjugate pairs,and partitions 3 + 1 and 2 + 1 + 1 are conjugate of each other. Of particular interest is the partition 2 + 2, which hasitself as conjugate. Such a partition is said to be self-conjugate.[17]
Claim: The number of self-conjugate partitions is the same as the number of partitions with distinct odd parts.Proof (outline): The crucial observation is that every odd part can be "folded" in the middle to form a self-conjugatediagram:One can then obtain a bijection between the set of partitions with distinct odd parts and the set of self-conjugatepartitions, as illustrated by the following example:
2.4.2 Odd parts and distinct parts
Among the 22 partitions of the number 8, there are 6 that contain only odd parts:
• 7 + 1
• 5 + 3
• 5 + 1 + 1 + 1
• 3 + 3 + 1 + 1
• 3 + 1 + 1 + 1 + 1 + 1
• 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
Alternatively, we could count partitions in which no number occurs more than once. If we count the partitions of 8with distinct parts, we also obtain 6:
• 8
• 7 + 1
12 CHAPTER 2. PARTITION (NUMBER THEORY)
• 6 + 2• 5 + 3• 5 + 2 + 1• 4 + 3 + 1
For all positive numbers the number of partitions with odd parts equals the number of partitions with distinct parts.[18]This result was proved by Leonhard Euler in 1748[19] and is a special case of Glaisher’s theorem.For every type of restricted partition there is a corresponding function for the number of partitions satisfying the givenrestriction. An important example is q(n), the number of partitions of n into distinct parts.[20] The first few values ofq(n) are (starting with q(0)=1):
1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, … (sequence A000009 in OEIS).
The generating function for q(n) (partitions into distinct parts) is given by[21]
∞∑n=0
q(n)xn =∞∏k=1
(1 + xk) =∞∏k=1
1
1− x2k−1.
The second product can be written ϕ(x2) / ϕ(x) where ϕ is Euler’s function; the pentagonal number theorem can beapplied to this as well giving a recurrence for q:[22]
q(k) = ak + q(k − 1) + q(k − 2) − q(k − 5) − q(k − 7) + q(k − 12) + q(k − 15) − q(k − 22) − ...
where ak is (−1)m if k = 3m2 − m for some integer m and is 0 otherwise.
2.4.3 Restricted part size or number of parts
Using the same conjugation trick as above, one may show that the number pk(n) of partitions of n into exactly kparts is equal to the number of partitions of n in which the largest part has size k.[23] The function pk(n) satisfies therecurrence
pk(n) = pk(n − k) + pk ₋ ₁(n − 1)
with initial values p0(0) = 1 and pk(n) = 0 if n ≤ 0 or k ≤ 0.One possible generating function for such partitions, taking k fixed and n variable, is
∑n≥0
pk(n)xn = xk ·
k∏i=1
1
1− xi.
More generally, if T is a set of positive integers then the number of partitions of n, all of whose parts belong to T,has generating function
∏t∈T
(1− xt)−1.
This can be used to solve change-making problems (where the set T specifies the available coins). As two particularcases, one has that the number of partitions of n in which all parts are 1 or 2 (or, equivalently, the number of partitionsof n into 1 or 2 parts) is
⌊n2+ 1⌋,
and the number of partitions of n in which all parts are 1, 2 or 3 (or, equivalently, the number of partitions of n intoat most three parts) is the nearest integer to (n + 3)2 / 12.[24]
2.5. RANK AND DURFEE SQUARE 13
Asymptotics
The asymptotic expression for p(n) implies that
log p(n) ∼ C√n as n → ∞
where C = π√
23 .[25]
If A is a set of natural numbers, we let pA(n) denote the number of partitions of n into elements of A. If A possessespositive natural density α then
log pA(n) ∼ C√αn
and conversely if this asymptotic property holds for pA(n) then A has natural density α.[26] This result was stated,with a sketch of proof, by Erdős in 1942.[11][27]
If A is a finite set, this analysis does not apply (the density of a finite set is zero). If A has k elements whose greatestcommon divisor is 1, then[28]
pA(n) =
(∏a∈A
a−1
)· nk−1
(k − 1)!+O(nk−2).
2.4.4 Partitions in a rectangle and Gaussian binomial coefficients
Main article: Gaussian binomial coefficient
One may also simultaneously limit the number and size of the parts. Let p(N, M; n) denote the number of partitionsof n with at most M parts, each of size at most N. Equivalently, these are the partitions whose Young diagram fitsinside an M × N rectangle. There is a recurrence relation
p(N,M ;n) = p(N,M − 1;n) + p(N − 1,M ;n−M)
obtained by observing that p(N,M ;n) − p(N,M − 1;n) counts the partitions of n into exactly M parts of size atmost N, and subtracting 1 from each part of such a partitions yields a partition of n − M into at most M parts.[29]
The Gaussian binomial coefficient is defined as:
(k + ℓ
ℓ
)q
=
(k + ℓ
k
)q
=
∏k+ℓj=1(1− qj)∏k
j=1(1− qj)∏ℓ
j=1(1− qj).
The Gaussian binomial coefficient is related to the generating function of p(N, M; n) by the equality
MN∑n=0
p(N,M ;n)qn =
(M +N
M
)q
.
2.5 Rank and Durfee square
Main article: Durfee square
14 CHAPTER 2. PARTITION (NUMBER THEORY)
The rank of a partition is the largest number k such that the partition contains at least k parts of size at least k. Forexample, the partition 4 + 3 + 3 + 2 + 1 + 1 has rank 3 because it contains 3 parts that are ≥ 3, but does not contain4 parts that are ≥ 4. In the Ferrers diagram or Young diagram of a partition of rank r, the r × r square of entries inthe upper-left is known as the Durfee square:
The Durfee square has applications within combinatorics in the proofs of various partition identities.[30] It also hassome practical significance in the form of the h-index.
2.6 Young’s lattice
Main article: Young’s lattice
There is a natural partial order on partitions given by inclusion of Young diagrams. This partially ordered set isknown as Young’s lattice. The lattice was originally defined in the context of representation theory, where it is used todescribe of the irreducible representations of symmetric groups Sn for all n, together with their branching properties,in characteristic zero. It also has received significant study for its purely combinatorial properties; notably, it is themotivating example of a differential poset.
2.7 Algorithm
The partition function is inherently recursive in nature since the results of smaller numbers appear as components inthe result of a larger number.Let p(n,m) be the number of partitions of n using only positive integers that are less than or equal to m.It may be seen that p(n) = p(n,n), and also p(n,m) = p(n,n) = p(n) for m > n.Therefore:
p(n, n) =
m∑k=1
p(n− k, k)
This recursive summation formula can be implemented with table lookups or memoization to remove duplicatedeffort.
2.8 See also• Rank of a partition, a different notion of rank
• Crank of a partition
• Dominance order
• Factorization
• Integer factorization
• Partition of a set
• Stars and bars (combinatorics)
• Plane partition
• Polite number, defined by partitions into consecutive integers
2.9. NOTES 15
• Multiplicative partition
• Twelvefold way
• Ewens’s sampling formula
• Faà di Bruno’s formula
• Multipartition
• Newton’s identities
• Leibniz’s distribution table for integer partitions
• Smallest-parts function
• A Goldbach partition is the partition of an even number into primes (see Goldbach’s conjecture)
• Kostant’s partition function
2.9 Notes[1] Andrews (1976) p.199
[2] Josuat-Vergès, Matthieu (2010), “Bijections between pattern-avoiding fillings of Young diagrams”, Journal of Combinato-rial Theory, Series A 117 (8): 1218–1230, arXiv:0801.4928, doi:10.1016/j.jcta.2010.03.006, MR 2677686.
[3] "Sloane’s A070177 ", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
[4] http://primes.utm.edu/top20/page.php?id=54
[5] Abramowitz and Stegun p. 825, 24.2.1 eq. I(B)
[6] Hardy and Wright (2008) Theorem 359, p.380
[7] Berndt and Ono, “Ramanujan’s Unpublished Manuscript on the Partition and Tau Functions with Proofs and Commentary”
[8] Ono (2004) p.87
[9] Hardy and Wright (2008) Theorems 360,361, p.380
[10] Andrews (1976) p.69
[11] Erdős, Pál (1942). “On an elementary proof of some asymptotic formulas in the theory of partitions”. Ann. Math. (2) 43:437–450. doi:10.2307/1968802. Zbl 0061.07905.
[12] Nathanson (2000) p.456
[13] F. Johansson, Efficient implementation of the Hardy-Ramanujan-Rademacher formula, LMS Journal of Computation andMathematics 15 (2012), 341-359.
[14] Fredrik Johansson (March 2, 2014). “New partition function record: p(1020) computed”.
[15] Alder, Henry L. (1969). “Partition identities - from Euler to the present”. Amer. Math. Monthly 76: 733–746.
[16] Hardy and Wright (2008) p.362
[17] Hardy and Wright (2008) p.368
[18] Hardy and Wright (2008) p.365
[19] Andrews, George E. Number Theory. W. B. Saunders Company, Philadelphia, 1971. Dover edition, page 149–150.
[20] Notation follows Abramowitz and Stegun p. 825
[21] Abramowitz and Stegun p. 825, 24.2.2 eq. I(B)
[22] Abramowitz and Stegun p. 826, 24.2.2 eq. II(A)
16 CHAPTER 2. PARTITION (NUMBER THEORY)
[23] Here the notation follows that of Stanley (1997), Section 1.
[24] Hardy, G.H. Some Famous Problems of the Theory of Numbers. Clarendon Press, 1920.
[25] Andrews (1976) pp70,97
[26] Nathanson (2000) pp.475–485
[27] Nathanson (2000) p.495
[28] Nathanson (2000) p.458–464
[29] Andrews (1976) pp.33-34
[30] see, e.g., Stanley (1997), p. 58
2.10 References
• Ahlgren, Scott; Ono, Ken (2001). “Congruence properties for the partition function” (PDF). Proceedings ofthe National Academy of Sciences 98 (23): 12882–12884. doi:10.1073/pnas.191488598. MR 1862931.
• George E. Andrews, The Theory of Partitions (1976), Cambridge University Press. ISBN 0-521-63766-X .
• Apostol, Tom M. (1990) [1976]. Modular functions and Dirichlet series in number theory. Graduate Texts inMathematics 41 (2nd ed.). New York etc.: Springer-Verlag. ISBN 0-387-97127-0. Zbl 0697.10023. (Seechapter 5 for a modern pedagogical intro to Rademacher’s formula).
• Hardy, G. H.; Wright, E. M. (2008) [1938]. An Introduction to the Theory of Numbers. Revised by D. R.Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.). Oxford: Oxford University Press.ISBN 978-0-19-921986-5. MR 2445243. Zbl 1159.11001.
• Lehmer, D. H. (1939). “On the remainder and convergence of the series for the partition function”. Trans.Amer. Math. Soc. 46: 362–373. doi:10.1090/S0002-9947-1939-0000410-9. MR 0000410. Zbl 0022.20401.Provides the main formula (no derivatives), remainder, and older form for A (n).)
• Gupta, Gwyther, Miller, Roy. Soc. Math. Tables, vol 4, Tables of partitions, (1962) (Has text, nearly completebibliography, but they (and Abramowitz) missed the Selberg formula for Ak(n), which is in Whiteman.)
• Macdonald, Ian G. (1979). Symmetric functions and Hall polynomials. Oxford Mathematical Monographs.Oxford University Press. ISBN 0-19-853530-9. Zbl 0487.20007. (See section I.1)
• Nathanson, M.B. (2000). Elementary Methods in Number Theory. Graduate Texts in Mathematics 195.Springer-Verlag. ISBN 0-387-98912-9. Zbl 0953.11002.
• Ono, Ken (2000). “Distribution of the partition function modulo m". Ann. of Math. 151 (1): 293–307.doi:10.2307/121118. MR 1745012. Zbl 0984.11050.
• Ono, Ken (2004). The web of modularity: arithmetic of the coefficients of modular forms and q-series. CBMSRegional Conference Series in Mathematics 102. Providence, RI: American Mathematical Society. ISBN0-8218-3368-5. Zbl 1119.11026.
• Sautoy, Marcus Du. The Music of the Primes. New York: Perennial-HarperCollins, 2003.
• Richard P. Stanley, Enumerative Combinatorics, Volumes 1 and 2. Cambridge University Press, 1999 ISBN0-521-56069-1
• Whiteman, A. L. (1956). A sum connected with the series for the partition function. Pacific Journal of Math. 6.pp. 159–176. Zbl 0071.04004. (Provides the Selberg formula. The older form is the finite Fourier expansionof Selberg.)
• Hans Rademacher, Collected Papers of Hans Rademacher, (1974) MIT Press; v II, p 100–107, 108–122, 460–475.
2.11. EXTERNAL LINKS 17
• Miklós Bóna (2002). A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory.World Scientific Publishing. ISBN 981-02-4900-4. (qn elementary introduction to the topic of integer parti-tion, including a discussion of Ferrers graphs)
• George E. Andrews, Kimmo Eriksson (2004). Integer Partitions. Cambridge University Press. ISBN 0-521-60090-1.
• 'A Disappearing Number', devised piece by Complicite, mention Ramanujan’s work on the Partition Function,2007
2.11 External links• Hazewinkel, Michiel, ed. (2001), “Partition”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Partition and composition calculator
• First 4096 values of the partition function
• An algorithm to compute the partition function
• Weisstein, Eric W., “Partition”, MathWorld.
• Weisstein, Eric W., “Partition Function P”, MathWorld.
• Pieces of Number from Science News Online
• Lectures on Integer Partitions by Herbert S. Wilf
• Counting with partitions with reference tables to the On-Line Encyclopedia of Integer Sequences
• Integer partitions entry in the FindStat database
• Integer::Partition Perl module from CPAN
• Fast Algorithms For Generating Integer Partitions
• Generating All Partitions: A Comparison Of Two Encodings
• Amanda Folsom, Zachary A. Kent, and Ken Ono, l-adic properties of the partition function. In press.
• Jan Hendrik Bruinier and Ken Ono, An algebraic formula for the partition function. In press.
Chapter 3
Partition of a set
For the partition calculus of sets, see infinitary combinatorics.In mathematics, a partition of a set is a grouping of the set’s elements into non-empty subsets, in such a way that
A set of stamps partitioned into bundles: No stamp is in two bundles, no bundle is empty, and every stamp is in a bundle.
every element is included in one and only one of the subsets.
18
3.1. DEFINITION 19
3.1 Definition
A partition of a set X is a set of nonempty subsets of X such that every element x in X is in exactly one of thesesubsets[2] (i.e., X is a disjoint union of the subsets).Equivalently, a family of sets P is a partition of X if and only if all of the following conditions hold:[3]
1. P does not contain the empty set.2. The union of the sets in P is equal to X. (The sets in P are said to cover X.)3. The intersection of any two distinct sets in P is empty. (We say the elements of P are pairwise disjoint.)
In mathematical notation, these conditions can be represented as
1. ∅ /∈ P
2.∪
A∈P A = X
3. if A,B ∈ P and A = B then A ∩B = ∅ ,
where ∅ is the empty set.The sets in P are called the blocks, parts or cells of the partition.[4]
The rank of P is |X| − |P|, if X is finite.
3.2 Examples• Every singleton set {x} has exactly one partition, namely { {x} }.• The empty set has exactly one partition, namely the empty set.• For any nonempty set X, P = {X} is a partition of X, called the trivial partition.• For any non-empty proper subset A of a set U, the set A together with its complement form a partition of U,namely, {A, U−A}.
• The set { 1, 2, 3 } has these five partitions:• { {1}, {2}, {3} }, sometimes written 1|2|3.• { {1, 2}, {3} }, or 12|3.• { {1, 3}, {2} }, or 13|2.• { {1}, {2, 3} }, or 1|23.• { {1, 2, 3} }, or 123 (in contexts where there will be no confusion with the number).
• The following are not partitions of { 1, 2, 3 }:• { {}, {1, 3}, {2} } is not a partition (of any set) because one of its elements is the empty set.• { {1, 2}, {2, 3} } is not a partition (of any set) because the element 2 is contained in more than one block.• { {1}, {2} } is not a partition of {1, 2, 3} because none of its blocks contains 3; however, it is a partitionof {1, 2}.
3.3 Partitions and equivalence relations
For any equivalence relation on a set X, the set of its equivalence classes is a partition of X. Conversely, from anypartition P of X, we can define an equivalence relation on X by setting x ~ y precisely when x and y are in the samepart in P. Thus the notions of equivalence relation and partition are essentially equivalent.[5]
The axiom of choice guarantees for any partition of a set X the existence of a subset of X containing exactly oneelement from each part of the partition. This implies that given an equivalence relation on a set one can select acanonical representative element from every equivalence class.
20 CHAPTER 3. PARTITION OF A SET
3.4 Refinement of partitions
A partition α of a set X is a refinement of a partition ρ of X—and we say that α is finer than ρ and that ρ is coarserthan α—if every element of α is a subset of some element of ρ. Informally, this means that α is a further fragmentationof ρ. In that case, it is written that α ≤ ρ.This finer-than relation on the set of partitions of X is a partial order (so the notation "≤" is appropriate). Each setof elements has a least upper bound and a greatest lower bound, so that it forms a lattice, and more specifically (forpartitions of a finite set) it is a geometric lattice.[6] The partition lattice of a 4-element set has 15 elements and isdepicted in the Hasse diagram on the left.Based on the cryptomorphism between geometric lattices and matroids, this lattice of partitions of a finite set cor-responds to a matroid in which the base set of the matroid consists of the atoms of the lattice, the partitions withn − 2 singleton sets and one two-element set. These atomic partitions correspond one-for-one with the edges of acomplete graph. The matroid closure of a set of atomic partitions is the finest common coarsening of them all; ingraph-theoretic terms, it is the partition of the vertices of the complete graph into the connected components of thesubgraph formed by the given set of edges. In this way, the lattice of partitions corresponds to the graphic matroidof the complete graph.Another example illustrates the refining of partitions from the perspective of equivalence relations. If D is the set ofcards in a standard 52-card deck, the same-color-as relation on D – which can be denoted ~C – has two equivalenceclasses: the sets {red cards} and {black cards}. The 2-part partition corresponding to ~C has a refinement that yieldsthe same-suit-as relation ~S, which has the four equivalence classes {spades}, {diamonds}, {hearts}, and {clubs}.
3.5 Noncrossing partitions
A partition of the set N = {1, 2, ..., n} with corresponding equivalence relation ~ is noncrossing provided that forany two 'cells’ C1 and C2, either all the elements in C1 are < than all the elements in C2 or they are all > than all theelements in C2. In other words: given distinct numbers a, b, c in N, with a < b < c, if a ~ c (they both are in a cellcalled C), it follows that also a ~ b and b ~ c, that is b is also in C. The lattice of noncrossing partitions of a finite sethas recently taken on importance because of its role in free probability theory. These form a subset of the lattice ofall partitions, but not a sublattice, since the join operations of the two lattices do not agree.
3.6 Counting partitions
The total number of partitions of an n-element set is the Bell number Bn. The first several Bell numbers are B0 =1, B1 = 1, B2 = 2, B3 = 5, B4 = 15, B5 = 52, and B6 = 203 (sequence A000110 in OEIS). Bell numbers satisfy therecursion
Bn+1 =n∑
k=0
(n
k
)Bk
and have the exponential generating function
∞∑n=0
Bn
n!zn = ee
z−1.
The Bell numbers may also be computed using the Bell triangle in which the first value in each row is copied fromthe end of the previous row, and subsequent values are computed by adding the two numbers to the left and above leftof each position. The Bell numbers are repeated along both sides of this triangle. The numbers within the trianglecount partitions in which a given element is the largest singleton.The number of partitions of an n-element set into exactly k nonempty parts is the Stirling number of the second kindS(n, k).The number of noncrossing partitions of an n-element set is the Catalan number Cn, given by
3.7. SEE ALSO 21
Cn =1
n+ 1
(2n
n
).
3.7 See also• Exact cover
• Cluster analysis
• Weak ordering (ordered set partition)
• Equivalence relation
• Partial equivalence relation
• Partition refinement
• List of partition topics
• Lamination (topology)
• Rhyme schemes by set partition
3.8 Notes[1] Knuth, Donald E. (2013), “Two thousand years of combinatorics”, in Wilson, Robin; Watkins, John J., Combinatorics:
Ancient and Modern, Oxford University Press, pp. 7–37
[2] Naive Set Theory (1960). Halmos, Paul R. Springer. p. 28. ISBN 9780387900926.
[3] Lucas, John F. (1990). Introduction to Abstract Mathematics. Rowman & Littlefield. p. 187. ISBN 9780912675732.
[4] Brualdi, pp. 44–45
[5] Schechter, p. 54
[6] Birkhoff, Garrett (1995), Lattice Theory, Colloquium Publications 25 (3rd ed.), American Mathematical Society, p. 95,ISBN 9780821810255.
3.9 References• Brualdi, Richard A. (2004). Introductory Combinatorics (4th ed.). Pearson Prentice Hall. ISBN 0-13-100119-1.
• Schechter, Eric (1997). Handbook of Analysis and Its Foundations. Academic Press. ISBN 0-12-622760-8.
22 CHAPTER 3. PARTITION OF A SET
The 52 partitions of a set with 5 elements
3.9. REFERENCES 23
1 2 3 4 5 6
7 8 9 10 11 12
13 14 15 16 17 18
19 20 21 22 23 24
25 26 27 28 29 30
31 32 33 34 35 36
37 38 39 40 41 42
43 44 45 46 47 48
49 50 51 52 53 54
The traditional Japanese symbols for the chapters of the Tale of Genji are based on the 52 ways of partitioning five elements.[1]
24 CHAPTER 3. PARTITION OF A SET
Partitions of a 4-set ordered by refinement
Chapter 4
Partition of an interval
A partition of an interval being used in a Riemann sum. The partition itself is shown in grey at the bottom, with one subintervalindicated in red.
In mathematics, a partition of an interval [a, b] on the real line is a finite sequence x = ( xi ) of real numbers suchthat
26
4.1. REFINEMENT OF A PARTITION 27
a = x0 < x1 < x2 < ... < xn = b.
In other terms, a partition of a compact interval I is a strictly increasing sequence of numbers (belonging to theinterval I itself) starting from the initial point of I and arriving at the final point of I.Every interval of the form [xᵢ, xᵢ₊₁] is referred to as a sub-interval of the partition x.
4.1 Refinement of a partition
Another partition of the given interval, Q, is defined as a refinement of the partition, P, when it contains all thepoints of P and possibly some other points as well; the partition Q is said to be “finer” than P. Given two partitions,P and Q, one can always form their common refinement, denoted P ∨ Q, which consists of all the points of P andQ, re-numbered in order.[1]
4.2 Norm of a partition
The norm (or mesh) of the partition
x0 < x1 < x2 < ... < xn
is the length of the longest of these subintervals,[2][3] that is
max{ |xi − xi₋₁| : i = 1, ..., n }.
4.3 Applications
Partitions are used in the theory of the Riemann integral, the Riemann–Stieltjes integral and the regulated integral.Specifically, as finer partitions of a given interval are considered, their mesh approaches zero and the Riemann sumbased on a given partition approaches the Riemann integral.[4]
4.4 Tagged partitions
A tagged partition[5] is a partition of a given interval together with a finite sequence of numbers t0, ..., tn₋₁ subjectto the conditions that for each i,
xᵢ ≤ tᵢ ≤ xᵢ₊₁.
In other words, a tagged partition is a partition together with a distinguished point of every subinterval: its mesh isdefined in the same way as for an ordinary partition. It is possible to define a partial order on the set of all taggedpartitions by saying that one tagged partition is bigger than another if the bigger one is a refinement of the smallerone.Suppose that x0, . . . , xn together with t0, . . . , tn−1 is a tagged partition of [a, b] , and that y0, . . . , ym together withs0, . . . , sm−1 is another tagged partition of [a, b] . We say that y0, . . . , ym and s0, . . . , sm−1 together is a refinementof a tagged partition x0, . . . , xn together with t0, . . . , tn−1 if for each integer i with 0 ≤ i ≤ n , there is an integerr(i) such that xi = yr(i) and such that ti = sj for some j with r(i) ≤ j ≤ r(i + 1) − 1 . Said more simply, arefinement of a tagged partition takes the starting partition and adds more tags, but does not take any away.
4.5 See also• Regulated integral
28 CHAPTER 4. PARTITION OF AN INTERVAL
• Riemann integral
• Riemann–Stieltjes integral
• Partition of a set
4.6 References[1] Brannan, D.A. (2006). AFirst Course inMathematical Analysis. CambridgeUniversity Press. p. 262. ISBN9781139458955.
[2] Hijab, Omar (2011). Introduction to Calculus and Classical Analysis. Springer. p. 60. ISBN 9781441994882.
[3] Zorich, Vladimir A. (2004). Mathematical Analysis II. Springer. p. 108. ISBN 9783540406334.
[4] Limaye, Balmohan (2006). A Course in Calculus and Real Analysis. Springer. p. 213. ISBN 9780387364254.
[5] Dudley, Richard M. & Norvaiša, Rimas (2010). Concrete Functional Calculus. Springer. p. 2. ISBN 9781441969507.
4.7 Further reading• Gordon, Russell A. (1994). The integrals of Lebesgue, Denjoy, Perron, and Henstock. Graduate Studies inMathematics, 4. Providence, RI: American Mathematical Society. ISBN 0-8218-3805-9.
Chapter 5
Partition of unity
In mathematics, a partition of unity of a topological space X is a set R of continuous functions from X to the unitinterval [0,1] such that for every point, x ∈ X ,
• there is a neighbourhood of x where all but a finite number of the functions of R are 0, and
• the sum of all the function values at x is 1, i.e.,∑
ρ∈R ρ(x) = 1 .
0
1
A partition of unity of a circle with four functions. The circle is unrolled to a line segment (the bottom solid line) for graphingpurposes. The dashed line on top is the sum of the functions in the partition.
Partitions of unity are useful because they often allow one to extend local constructions to the whole space. They arealso important in the interpolation of data, in signal processing, and the theory of spline functions.
5.1 Existence
The existence of partitions of unity assumes two distinct forms:
1. Given any open cover {Ui}i∈I of a space, there exists a partition {ρi}i∈I indexed over the same set I such thatsupp ρi⊆Ui. Such a partition is said to be subordinate to the open cover {Ui}i.
2. Given any open cover {Ui}i∈I of a space, there exists a partition {ρj}j∈J indexed over a possibly distinct indexset J such that each ρj has compact support and for each j∈J, supp ρj⊆Ui for some i∈I.
Thus one chooses either to have the supports indexed by the open cover, or compact supports. If the space is compact,then there exist partitions satisfying both requirements.A finite open cover always has a continuous partition of unity subordinated to it, provided the space is locally compactand Hausdorff.[1] Paracompactness of the space is a necessary condition to guarantee the existence of a partition ofunity subordinate to any open cover. Depending on the category which the space belongs to, it may also be a sufficientcondition.[2] The construction uses mollifiers (bump functions), which exist in continuous and smooth manifolds, butnot in analytic manifolds. Thus for an open cover of an analytic manifold, an analytic partition of unity subordinateto that open cover generally does not exist. See analytic continuation.If R and S are partitions of unity for spaces X and Y, respectively, then the set of all pairwise products { ρσ : ρ ∈R ∧ σ ∈ S } is a partition of unity for the cartesian product space X×Y.
29
30 CHAPTER 5. PARTITION OF UNITY
5.2 Variant definitions
Sometimes a less restrictive definition is used: the sum of all the function values at a particular point is only requiredto be positive, rather than 1, for each point in the space. However, given such a set of functions, one can obtain apartition of unity in the strict sense by dividing every function by the sum of all functions (which is defined, since atany point it has only a finite number of terms).
5.3 Applications
A partition of unity can be used to define the integral (with respect to a volume form) of a function defined over amanifold: One first defines the integral of a function whose support is contained in a single coordinate patch of themanifold; then one uses a partition of unity to define the integral of an arbitrary function; finally one shows that thedefinition is independent of the chosen partition of unity.A partition of unity can be used to show the existence of a Riemannian metric on an arbitrary manifold.Method of steepest descent employs a partition of unity to construct asymptotics of integrals.Linkwitz–Riley filter is an example of practical implementation of partition of unity to separate input signal into twooutput signals containing only high- or low-frequency components.The Bernstein polynomials of a fixed degree m are a family of m+1 linearly independent polynomials that are apartition of unity for the unit interval [0, 1] .
5.4 See also• Smoothness § Smooth partitions of unity
• Gluing axiom
• Fine sheaf
5.5 References[1] Rudin, Walter (1987). Real and complex analysis (3rd ed.). New York: McGraw-Hill. p. 40. ISBN 0-07-054234-1.
[2] Aliprantis, Charalambos D.; Border, Kim C. (2007). Infinite dimensional analysis: a hitchhiker’s guide (3rd ed.). Berlin:Springer. p. 716. ISBN 978-3-540-32696-0.
• Tu, LoringW. (2011),An introduction to manifolds, Universitext (2nd ed.), Berlin, NewYork: Springer-Verlag,doi:10.1007/978-1-4419-7400-6, ISBN 978-1-4419-7399-3, see chapter 13
5.6 External links• General information on partition of unity at [Mathworld]
• Applications of a partition of unity at [Planet Math]
Chapter 6
Partition regularity
In combinatorics, a branch of mathematics, partition regularity is one notion of largeness for a collection of sets.Given a set X , a collection of subsets S ⊂ P(X) is called partition regular if every set A in the collection has theproperty that, no matter how A is partitioned into finitely many subsets, at least one of the subsets will also belong tothe collection. That is, for any A ∈ S , and any finite partition A = C1 ∪ C2 ∪ · · · ∪ Cn , there exists an i ≤ n, suchthat Ci belongs to S . Ramsey theory is sometimes characterized as the study of which collections S are partitionregular.
6.1 Examples• the collection of all infinite subsets of an infinite setX is a prototypical example. In this case partition regularityasserts that every finite partition of an infinite set has an infinite cell (i.e. the infinite pigeonhole principle.)
• sets with positive upper density inN : the upper density d(A) ofA ⊂ N is defined as d(A) = lim supn→∞|{1,2,...,n}∩A|
n .
• For any ultrafilter U on a setX , U is partition regular. If U ∋ A =∪n
1 Ci , then for exactly one i is Ci ∈ U .
• sets of recurrence: a set R of integers is called a set of recurrence if for any measure preserving transformationT of the probability space (Ω, β, μ) and A ∈ β of positive measure there is a nonzero n ∈ R so thatµ(A ∩ TnA) > 0 .
• Call a subset of natural numbers a.p.-rich if it contains arbitrarily long arithmetic progressions. Then thecollection of a.p.-rich subsets is partition regular (Van der Waerden, 1927).
• Let [A]n be the set of all n-subsets of A ⊂ N . Let Sn =∪
A⊂N[A]n . For each n, Sn is partition regular.
(Ramsey, 1930).
• For each infinite cardinal κ , the collection of stationary sets of κ is partition regular. More is true: if S isstationary and S =
∪α<λ Sα for some λ < κ , then some Sα is stationary.
• the collection of∆ -sets: A ⊂ N is a∆ -set ifA contains the set of differences {sm−sn : m,n ∈ N, n < m}for some sequence ⟨sn⟩ωn=1 .
• the set of barriers on N : call a collection B of finite subsets of N a barrier if:
• ∀X,Y ∈ B, X ⊂ Y and• for all infinite I ⊂ ∪B , there is someX ∈ B such that the elements of X are the smallest elements of I;i.e. X ⊂ I and ∀i ∈ I \X,∀x ∈ X,x < i .
This generalizes Ramsey’s theorem, as each [A]n is a barrier. (Nash-Williams, 1965)
31
32 CHAPTER 6. PARTITION REGULARITY
• finite products of infinite trees (Halpern–Läuchli, 1966)
• piecewise syndetic sets (Brown, 1968)
• Call a subset of natural numbers i.p.-rich if it contains arbitrarily large finite sets together with all their finitesums. Then the collection of i.p.-rich subsets is partition regular (Folkman–Rado–Sanders, 1968).
• (m, p, c)-sets (Deuber, 1973)
• IP sets (Hindman, 1974, see also Hindman, Strauss, 1998)
• MTk sets for each k, i.e. k-tuples of finite sums (Milliken–Taylor, 1975)
• central sets; i.e. the members of any minimal idempotent in βN , the Stone–Čech compactification of theintegers. (Furstenberg, 1981, see also Hindman, Strauss, 1998)
6.2 References1. Vitaly Bergelson, N. Hindman Partition regular structures contained in large sets are abundant J. Comb. Theory
(Series A) 93 (2001), 18–36.
2. T. Brown, An interesting combinatorial method in the theory of locally finite semigroups, Pacific J. Math. 36,no. 2 (1971), 285–289.
3. W. Deuber, Mathematische Zeitschrift 133, (1973) 109–123
4. N. Hindman, Finite sums from sequences within cells of a partition of N, J. Combinatorial Theory (Series A)17 (1974) 1–11.
5. C.St.J.A. Nash-Williams, On well-quasi-ordering transfinite sequences, Proc. Camb. Phil. Soc. 61 (1965),33–39.
6. N. Hindman, D. Strauss, Algebra in the Stone–Čech compactification, De Gruyter, 1998
7. J.Sanders, A Generalization of Schur’s Theorem, Doctoral Dissertation, Yale University, 1968.
6.3. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 33
6.3 Text and image sources, contributors, and licenses
6.3.1 Text• Multiplicative partition Source: https://en.wikipedia.org/wiki/Multiplicative_partition?oldid=504695752 Contributors: Michael Hardy,
Melchoir, Richard L. Peterson, CRGreathouse, BackwardsBoy, David Eppstein, R'n'B, Ixionid, Alksentrs, Bluebusy, Erik9bot, Robo37,Citation bot 1, Kirontanvir11, ChrisGualtieri and Anonymous: 7
• Partition (number theory) Source: https://en.wikipedia.org/wiki/Partition_(number_theory)?oldid=706547637 Contributors: Tarquin,Nonenmac, Stevertigo, Michael Hardy, Pwlfong, Wshun, Eliasen, Kku, Ixfd64, TakuyaMurata, GTBacchus, Charles Matthews, Phys,McKay, Fredrik, Gandalf61, Merovingian, Henrygb, Robinh, Giftlite, Chinasaur, Jason Quinn, Macrakis, HorsePunchKid, Almit39,4pq1injbok, SamDerbyshire, Bender235, El C, Crisófilax, Burn, OlegAlexandrov, Joriki, Simetrical, Linas, Thehebrewhammer, Rjwilmsi,Jivecat, Jwmcleod, Fivemack, Maxal, Hillman, Michael Slone, Anomalocaris, Bruguiea, Tetracube, Redgolpe, Ilmari Karonen, Smack-Bot, RDBury, Rōnin, Gutworth, Timothy Clemans, Mhym, Loopology, Lambiam, Richard L. Peterson, Shoeofdeath, JRSpriggs, Timrem,Ylloh, CRGreathouse, Thijs!bot, Headbomb, Wang ty87916, JoaquinFerrero, Hannes Eder, GromXXVII, Arch dude, David Eppstein,Miaers, CommonsDelinker, Lantonov, Krishnachandranvn, Daniel5Ko, Policron, Milogardner, Philip Trueman, TXiKiBoT, Mathman99,Anchor Link Bot, Rumping, Justin W Smith, EGetzler, Mild Bill Hiccup, FractalFusion, Niceguyedc, Sambrow, DragonBot, Kingvashy,Watchduck, Bender2k14, JNLII, Gciriani, Marc van Leeuwen, Jed 20012, Khunglongcon, Addbot, Zdaugherty, Download, Lightbot,,אבינעם Bluebusy, Legobot, Yobot, Ptbotgourou, , Citation bot, Obersachsebot, Xqbot, Coretheapple, RibotBOT, FrescoBot, Robo37,GerardSchildberger, FoxBot, Xnn, The tree stump, EmausBot, Slawekb, R. J. Mathar, Suslindisambiguator, Vladimirdx, Ebehn, Clue-Bot NG, SeekingAnswers, DependableSkeleton, KiruJiwak, Mesoderm,Matt.mawson, Joel B. Lewis, Archimedes100, Partedit, BG19bot,Karun3kumar, Googleisdik, NereusAJ,Mpsimo, Alexjbest, Deltahedron, Spectral sequence, Haphaeu, Peter13542, Vesoto, Ssuben, Lasy-dler and Anonymous: 114
• Partition of a set Source: https://en.wikipedia.org/wiki/Partition_of_a_set?oldid=706691220 Contributors: AxelBoldt, Tomo, Patrick,Michael Hardy, Wshun, Kku, Revolver, Charles Matthews, Zero0000, Robbot, MathMartin, Ruakh, Tobias Bergemann, Giftlite, Smjg,Arved, Fropuff, Gubbubu, Mennucc, Sam Hocevar, Tsemii, TedPavlic, Paul August, Zaslav, Elwikipedista~enwiki, El C, PhilHibbs,Corvi42, Oleg Alexandrov, Stemonitis, Bobrayner, Kelly Martin, Linas, MFH, Mayumashu, Salix alba, R.e.b., FlaBot, Mathbot, Yurik-Bot, Laurentius, Gaius Cornelius, StevenL, Pred, Finell, Capitalist, That Guy, From That Show!, Adam majewski, Mcld, Mhss, Tsca.bot,Mhym, Armend, JonAwbrey, RobZako, CRGreathouse, 345Kai, Sopoforic, Escarbot, Magioladitis, Jiejunkong, David Eppstein, Lantonov,Elenseel, TXiKiBoT, Anonymous Dissident, PaulTanenbaum, Jamelan, Skippydo, PipepBot, DragonBot, Watchduck, Addbot, AkhtaBot,Legobot, Luckas-bot, Yobot, Bunnyhop11, Calle, AnomieBOT, ArthurBot, Stereospan, MastiBot, EmausBot, MartinThoma, ZéroBot,D.Lazard, Orange Suede Sofa, ClueBot NG, Mesoderm, Helpful Pixie Bot, BG19bot, BattyBot, MinatureCookie, Mark viking, Cepphus,BethNaught and Anonymous: 48
• Partition of an interval Source: https://en.wikipedia.org/wiki/Partition_of_an_interval?oldid=675185797 Contributors: Michael Hardy,Charles Matthews, Giftlite, Ncik~enwiki, Oleg Alexandrov, Mindmatrix, Uncle G, Flamingspinach, YurikBot, SpuriousQ, Kompik,Lt-wiki-bot, SmackBot, Diegotorquemada, WISo, Hypergeek14, Sullivan.t.j, Addbot, Yobot, Calle, Citation bot, Nicolas Perrault III,DixonDBot, Mesoderm, Davidcarfi, Solomon7968, YFdyh-bot, Brirush, Mark viking, Mrpalermo and Anonymous: 10
• Partition of unity Source: https://en.wikipedia.org/wiki/Partition_of_unity?oldid=689944180 Contributors: Mav, Toby~enwiki, Dys-prosia, Giftlite, BenFrantzDale, Lethe, Fropuff, Dratman, Jorge Stolfi, ArnoldReinhold, Oleg Alexandrov, MFH, YurikBot, Trovatore,SmackBot, MisterHand, Mhym, Dreadstar, N2e, Headbomb, Albmont, Jakob.scholbach, Joedalion, Marcosaedro, YouRang?, MystBot,Addbot, Topology Expert, Numbo3-bot, Ht686rg90, Txebixev, Point-set topologist, Erik9bot, Rausch, Mickeyjetson428, Boriaj, Batty-Bot, Khazar2, Denysbondar and Anonymous: 11
• Partition regularity Source: https://en.wikipedia.org/wiki/Partition_regularity?oldid=607156121 Contributors: Michael Hardy, CharlesMatthews, Altenmann, Tobias Bergemann, Giftlite, Redquark, Greg321, JdH, Landonproctor, Ksoileau, FizzyP, Sullivan.t.j, David Epp-stein, Darnedfrenchman, Addbot, Yobot, Kiefer.Wolfowitz, Xnn, WikitanvirBot and Anonymous: 6
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34 CHAPTER 6. PARTITION REGULARITY
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