Combinatorial interpretations for a class of algebraic equations and

uniform partitions

Speaker: Yeong-Nan Yeh

Institute of mathemetics, Academia sinica

Aug. 21, 2012

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Catalan paths• An n-Catalan path is a lattice path from (0,0)

to (2n,0) in the first quadrant consisting of up-step (1,1) and down-step (1,-1) .

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Catanlan number

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, … ,

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2)(1)(

:equation Algebaric

zzCzC

0

)(:function Generatingn

nn zczC

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Motzkin paths• An n-Motizkin path is a lattice path from (0,0)

to (n,0) in the first quadrant consisting of up-step (1,1), level-step (1,0) and down-step (1,-1).

Motzkin number:1, 1, 2, 4, 9, 21, 51, 127, 323, 835, … ,

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22 )()(1)(

:equation Algebaric

zMzzzMzM

0

)(:function Generatingn

nn zmzM

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Combinatorial structure Generating function f(z) Algebaric equation

Catalan path:(1,1),(1,-1) in the first quadrant

C(z) C(z)=1+z[C(z)]2

=1+zC(z)·C(z)

Motzkin path:(1,1),(1,-1),(1,0) in the first quadrant

M(z) M(z)=1+zM(z)+z2[M(z)]2

=1+zM(z)·[1+zM(z)]

??? ??f(z)

Given an algebaric equation for arbitrary polynomial F(z,y) , how to construct a combinatorial structure such that its generating function f(z) satisfies this equation?

),(1 yzFzyy

),(1 yzzyFy

))(,()(1)( zfzFzzfzf

第 8页

Lattice paths• A lattice path is a sequence

(x1,y1)(x2,y2)…(xk,yk) of vectors in the plane with (xi,yi)∈Z≥0×Z\{(0,0)}, where Z and Z≥0 are the sets of integers and nonnegative integers respectively.

第 9页

Weight of a lattice path• Let w be a function from Z≥0

×Z to R, where R is the set of real numbers.

• For any lattice path P=(x1,y1)(x2,y2)…(xk,yk) define the weight of P, denoted by w(P), as

k

iii yxwPw

1

),()(

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S-path and S-nonnegative path• Let S be a finite subset of Z≥0

×Z\{(0,0)}.

• An S-path is a lattice path (x1,y1)(x2,y2)…(xk,yk) with (xi,yi) S.∈

• An S-nonnegative path is an S-path in the first quadrant

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)).(,()(1

)]([1 Then

.)(function generating a Define .)( and

(n,0)endpoint with paths enonnegativ-S ofset thebe )(Let

.)1(1)10(

)},,2,1,,2,1)1,{()}1,0{(Let :Theorem

1 1

00

zfzFzzf

zfzaf(z)

zPwf(z)LSLL

SLL

aj,-i,w,w

mjriijS

ijr

i

m

jij

n LP

n

nn

nn

ij

n

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• A decomposition of a S-nonnegative path.

P=(0,1)P1(0,1)P2(0,1)P3…Pi-1(j,-i+1)Pi

w(0,1)=1,w(j,-i+1)=ai,j

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More general cases

• Let λ be a function from Z≥0 ×Z to Z≥0.

• For any lattice path P=(x1,y1)(x2,y2)…(xk,yk) define the λ-length of P, denoted by λ(P), as

k

iii yxP

1

),()(

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)).(,()(1

1

Then .function generating a Define

. and axis-on point a

at ending length - with paths enonnegativ-S ofset thebe )(Let

.1)1,( and,0)1,()1()1,()1(

Let .integers enonnegativ are and where

,)0,0( and )},,2,1,,2,1)1,{()}1,{(

Let :Theorem

1 1

0

1

zfzFzzf

[f(z)]zaf(z)

w(P)zf(z)

(S)LL(S)Lx

nSLL

a),-iw(wj, w,-ii-

SmjriiS

ijr

i

m

jij

LP

λ(P)

nn

nn

ijji

j

j

j

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Uniform partition• An n-Dyck path is a lattice path from (0,0) to (2n,0) in the

plane integer lattice Z×Z consisting of up-step (1,1) and down-step (1,-1).

The number of n-Dyck paths is

n

n2

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K.L. Chung, W. Feller, On fluctuations in-coin tossing, Proc. Natl. Acad. Sci. USA 35 (1949) 605-608

Chung-Feller theorem:The number of Dyck path of semi-length n with m up-steps under x-axis is the n-th Catalan number and independent on m.

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Uniform partition (An uniform partition for Dyck paths)

The number of up-steps (1,1) lying below x-axis

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0 0

,

0 0,

1

)()()(

have We

orem,Feller theChungby with ngSubstituti

)(

Let

n

n

k

knn

nkn

n

n

k

knkn

y

zCyzyCyzcy,zC

cc

yzcy,zC

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Lifted Motzkin paths• A lifted n-Motizkin path is a lattice path from (0,0)

to (n+1,1) in the plane integer lattice Z×Z consisting of up-step (1,1), level-step (1,0) and down-step (1,-1), which never passes below the line y=1 except (0,0).

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Free Lifted Motzkin paths• A free lifted n-Motizkin path is a lattice path from (0,0) to

(n+1,1) in the plane integer lattice Z×Z consisting of up-step (1,1), level-step (1,0) and down-step (1,-1).

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• The number of free lifted n-Motzkin path with m steps at the left of the rightmost lowest point is the n-th Motzkin number and independent on m.

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An uniform partition for free lifted Motzkin paths• Shapiro found an uniform partition for Motzkin path.

L. Shapiro, Some open questions about random walks, involutions, limiting distributions, and generating functions, Advances in Applied Math. 27 (2001), 585-596.

The number of steps at the left of the rightmost lowest point of a lattice path

Eu, Liu and Yeh proved this proposition.Eu, S. P. Liu, S. C. and Yeh, Y. N. Taylor expansions for Catalan and Motzkin numbers, Adv. Appl. Math. 29 (2002) 345-357

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0 0

,

0 0,

1

)()()(

have We

,with ngSubstituti

)(

Let

n

n

k

knn

nkn

n

n

k

knkn

y

zMyzyMyzmy,zM

mm

yzmy,zM

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Function of uniform partition type

• For any generating function f(x),the form

is called the function of uniform partition type for f(x).

1

)()(

y

zfyzyf

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Combinatorial structure

Generating function Combinatorial structure

Function of uniform partition

Catalan path C(z)

C(z)=1+z[C(z)]2

n-Dyck path:

Motzkin path M(z)

M(z)=1+zM(z)+z2[M(z)]2

lifted n-Motizkin path

??? f(z)

f(z)=1+yzF(y,f(z))

???

1

)()(),(

y

zCyzyCzyC

1

)()(),(

y

zMyzyMzyM

1

)()(),(

y

zfyzyfzyCF

Function of uniform partition type

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.)11(1)11(

and

)},,1,,2,1)1,1{()}1,1{(

set stepFix

ija,-iij,w,w

mijriiijS

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)).(,()(1

)]([1 Then

.)(

function generating a Define .)( and

(n,0)endpoint with paths enonnegativ-S ofset thebe )(Let

.)11(1)11(

)},,1,,2,1)1,1{()}1,1{(Let

:Lemma

1 1

0

0

zfzFzzf

zfzaf(z)

zPwf(z)

LSLL

SLL

a,-iij,w,w

mijriiijS

ijr

i

m

jij

n LP

n

nn

nn

ij

n

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• A decomposition of a S-nonnegative path.

P=(1,1)P1(1,1)P2(1,1)P3…Pi-1(j-i+1,-i+1)Pi

w(1,1)=1, w(j-i+1,-i+1)=ai,j

(j-i+1,-i+1)

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Function of uniform partition type

r

i

m

ij

i

k

kikkjjji

r

i

m

ij

ij

k

ikjji

r

i

m

ij

ijji

nn

yzfzfyza

zfyza

zyCF

y

zfyzyfzyCF

zfzazf

zfzf

1 1

1

0

1,

1 1 0,

1 1,

0n

)()(1

)(1

),(

Then .1

)()(),(Let

.)(1)(

equation algebaric following thesatisfies )( Suppose

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r

i

m

ij

i

k

kkikjjij

r

i

m

ij

iij

k

kjij

zfyzfyzazyH

zfyzazyG

1 1

1

0

1

1 1 0

)]([)]([1

1),(

)]([1),(Let

r

i

m

ij

i

k

kikkjjij

r

i

m

ij

ij

k

ikjij

yzfzfyza

zfyza

zyCF

1 1

1

0

1

1 1 0

)()(1

)(1

),(

Combinatorial interpretations for H(y,z) and G(y,z)?

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Combinatorial interpretation for H(y,z)

• Recall that an S-path is a lattice path P=(x1,y1)(x2,y2)…(xk,yk) with (xi,yi) S.∈

• Define the nonpositive length of P, denoted by nl(P), as the sum of x-coordinate of steps touching or going below x-axis.

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Combinatorial interpretation for H(y,z)• Define the nonpositive length of P, denoted

by nl(P), as the sum of x-coordinate of steps touching or going below x-axis,

nl(P)=1+1+1+2+1+1=7

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.)(,

Then .)( and

(n,0)endpoint with paths-S ofset thebe )(Let

.)11(1)11(

)},,1,,2,1)1,1{()}1,1{(Let

:Lemma

0

)(

0

n HP

nPnl

nn

nn

ij

n

zyPwz)H(y

HSHH

SHH

a,-iij,w,w

mijriiijS

Combinatorial interpretation for H(y,z)

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f(z)f(z)

f(z)

f(yz)

f(yz)f(yz)

H(y,z)

kkikjjij zfyzfyza )]([)]([ 1

r

i

m

ij

i

k

kkikjjij zfyzfyzazyHzyH

1 1

1

0

1 )]([)]([),(1),(

• A decomposition of a S-path.

第 35页

• A rooted S-nonnegative path is a pair [P;k] consisting of an S-nonnegative path P=(x1,y1)(x2,y2)…(xn,yn) with xn≥1 and a nonnegative integer k with 0≤ k≤ xn-1.

• For example, P=(1,1)(1,1)(1,-2)(1,0)(1,1)(1,1)(1,1)(1,-1)(2,-1).[P;0],[P;1] and [P;2] are rooted S-nonnegative path.

Combinatorial interpretation for G(y,z)

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.)(,

Then .)( and (n,0)endpoint with

paths enonnegativ-S rooted ofset thebe )(Let

.)11(1)11(

)},,1,,2,1)1,1{()}1,1{(Let

:Lemma

0 ];[

0

n GkP

nk

nn

nn

ij

n

zyPwz)G(y

GSGG

SGG

a,-iij,w,w

mijriiijS

Combinatorial interpretation for G(y,z)

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ikjij zfyza )]([

f(z)

f(z)f(z)

f(z)

k

• A decomposition of a rooted S-nonnegative path.

r

i

m

ij

iij

k

kjij zfyzazyG

1 1 0

)]([1),(

第 38页

• A lifted S-path is an S-path in the plane starting at (0,0) and ending at a point in the line y=1.

• A rooted lifted S-path is a pair [P;k] consisting of a lifted S-path P=(x1,y1)(x2,y2)…(xn,yn) with xn≥1 and a nonnegative integer k with 0≤ k≤ xn-1.

Combinatorial interpretation for CF(y,z)

第 39页

Combinatorial interpretation for CF(y,z)

.)(,F

Then .)( and 1,1)(nendpoint with

paths-S lifted rooted ofset thebe )(Let

.)11(1)11(

)},,1,,2,1)1,1{()}1,1{(Let

:Theorem

0 ];[

)(

0

n TkP

nkPnl

nn

nn

ij

n

zyPwz)(yC

TSTT

STT

a,-iij,w,w

mijriiijS

第 40页第 40页

The last step (1,1) from y=0 to y=1

(n+1-k,0)

A decomposition for an rooted lifted S-path

G(y,z)

H(y,z)

.)(

),(),(,F

0 ];[

)(

n TkP

nkPnl

n

zyPw

zyGzyHz)(yC

(n+1,1)

(0,0)

第 41页

More general cases

• Let λ be a function from Z≥0 ×Z to Z≥0.

• For any lattice path P=(x1,y1)(x2,y2)…(xk,yk) define the λ-length of P, denoted by λ(P), as

k

iii yxP

1

),()(

第 42页

• A λ-rooted lifted S-path is a pair [P;k] consisting of a lifted S-path P=(x1,y1)(x2,y2)…(xn,yn)

with λ(xn,yn)≥1 and a nonnegative integer k with 0≤

k≤ λ(xn,yn) -1.

第 43页

• For any S-path P=(x1,y1)(x2,y2)…(xn,yn), define the λ-nonpositive length of P, denoted by nlλ(P), as the sum of λ-length of steps touching or going below x-axis.

• For example, let λ(1,1)=0, λ(x,y)=x for any (x,y)≠ （ 1,1）

nlλ (P)=1+1+0+2+1+0=5

• For any rooted lifted S-path [P;k], define the rootedλ-nonpositive length of P as nlλ(P)+k.

第 44页n

,

];[

1)()(

1 10

j1

and 0any for on t independen

is Then .length enonpositiv rooted- and 1length -

with paths-S lifted rooted- of weightsof sum thebe Let (3)

.)(1

)()(,F

Then .paths-S lifted rooted ofset thebe (2)Let

)(1)(Then .)()(

function generating a Define axis.-on point aat ending

nlength - with paths enonnegativ-S ofset thebe Let (1)

.1)1(1,,1)(;)1()1(

)},,1,,2,1)1,{()}1,{(Let :Theorem

ffnkk

fkn

f

zyPwy

zfyzyfz)(yC

T

zfzazfzPwzf

x

L

j-i,-iαa,-iw,w

mijriiS

n,k

kn

n,k

TkP

PkPnl

r

i

m

ij

ijij

n

n LP

n

jiji-

j

n

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Thank you for your attention!