Post on 06-Jan-2016
description
Combinatorial interpretations for a class of algebraic equations and
uniform partitions
Speaker: Yeong-Nan Yeh
Institute of mathemetics, Academia sinica
Aug. 21, 2012
第 2页第 2页
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
第 5页第 5页
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
第 7页第 7页
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
),()(
第 10页
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
第 12页第 12页
• 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
第 13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 14页第 14页
)).(,()(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
第 15页
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
第 16页第 16页
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
第 19页第 19页
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).
第 20页第 20页
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
第 23页
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
第 24页
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
第 25页第 25页
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
第 27页第 27页
)).(,()(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
第 28页第 28页
• 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)
第 29页
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
第 30页第 30页
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.
第 32页第 32页
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
第 33页第 33页
.)(,
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)
第 34页第 34页
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)
第 36页第 36页
.)(,
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)
第 37页第 37页
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
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fkn
f
zyPwy
zfyzyfz)(yC
T
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x
L
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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!