combinatorial expansions for paths, chung-feller theorem and hankel matrix

Combinatorial Expansions for Paths, Chung-Feller Theorem

and Hankel Matrix

Speaker: Yeong-Nan Yeh

Institute of mathemetics, Academia sinica

2

Online

Part I. Functions of uniform-partition type Part II. Combinatorial interpretations for a class

of function equations Part III. Lattice paths and Fluctuation theory Part IV Paths with some avoiding sets shift

equivalence Part V. Addition formulas of polynomials and

Hankel determinants

3

Part I.

Functions of

uniform-partition type

4

Catalan paths

An n-Catalan path is a lattice path in the first quadrant starting at (0,0) and ending at (2n,0) with only two kinds of steps---up-step: U=(1,1) and down- step: D=(1,-1).

5

Catanlan number

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

6

Catanlan number

The Catalan sequence was first described in the 18th century by Leonhard Euler, who was interested in the number of different ways of dividing a polygon into triangles.

7

Eugène Charles Catalan (May 30, 1814 – February 14, 1894) was a French and Belgian mathematician.

The sequence is named after Eugène Charles Catalan, who discovered the connection to parenthesized expressions during his exploration of the Towers of Hanoi puzzle.

E.C. Catalan, Note surune equation aux di erences finies, J. Math.Pures Appl. ff3(1838), 508-515.

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

8

Catanlan number

The counting trick for Catalan words was found by D. André in 1887

D. André, Solution directe du problème résolu par M. Bertrand, Comptes Rendus de l’Académie des Sciences, Paris 105 (1887) 436–437.

9

Chung-Feller Theorem (The number of Dyck path of semi-length n with m nonpositive up-steps is the n-th Catalan number and independent on m.)

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

We say Chung-Feller theorem is an uniform

partition of up-down type.

10

The classical Chung-Feller theorem was proved by Macmahon.

MacMahon, P. A. Memoir on the theory of the partitions of numbers, Philos. Trans. Roy. Soc. London, Ser. A, 209 (1909), 153-175.

Chung and Feller reproved the theorem by analytic method.

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

A combinatorial proof.

Narayana, T. V. Cyclic permutation of lattice paths and the Chung-Feller theorem, Skand. Aktuarietidskr. (1967) 23-30

Eu, Liu and Yeh proved the Chung-Feller theorem by using the Taylor expansions of generating functions.

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

Eu, Fu and Yeh gave a strengthening of the Chung-Feller Theorem and a weighted version for schroder paths.

Eu, S. P. Fu, T. S. and Yeh, Y. N. Refined Chung-Feller theorems for lattice paths, J. Combin. Theory Ser. A 112 (2005) 143-162

11

Bijection proofs.

D. Callan, Pair them up! A visual approach to the Chung-Feller theorem, Coll. Math. J. 26(1995)196-198.R.I. Jewett, K. A. Ross, Random walk on Z, Coll. Math. J. 26(1995)196-198.

Mohanty’s book devotes an entire section to exploring the Chung-Feller theorem.

Mohanty, S. G. Lattice path counting and applications, NewYork : Academic Press, 1979.

Narayana's book introduced a refinement of this theorem.

T.V. Narayana, Lattice path combinatorics, with statistical applications,Toronto;Buffalo : University of Toronto Press, c1979.

Callan reviewed and compared combinatorial interpretations of three different expressions for the Catalan number by cycle method.

D. Callan, Why are these equal? http://www.stat.wisc.edu/~callan/notes/ Huq developed generalized versions of this theorem for lattice paths.

A. Huq, Generalized Chung-Feller Theorems for Lattice Paths(Thesis), http://arxiv.org/abs/0907.3254

12

W.J. Woan, Uniform partitions of lattice paths and Chung-Feller Generalizations, Amer. Math. Monthly 108(2001) 556-559.

Another uniform partition for Dyck paths

The number of up-steps at the left of the rightmost lowest point of a dyck path

We say this uniform partition

is of left-right type.

13

Motzkin paths

An n-Motizkin path is a lattice path in the first quadrant starting at (0,0) and ending at (n,0) with only two kinds of steps---level-step: (1,0), up-step: U=(1,1) and down- step: D=(1,-1).

14

An uniform partition for 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

This uniform partition is of left-right type.

Eu, Liu and Yeh proved this proposition.


15

Another uniform partition of up-down type for Motzkin paths.

The number of steps touching x-axis and under x-axis

第 16页

Our main results1. Eu, Liu and Yeh proved the Chung-Feller theorem by using the Taylor

expansions of generating functions.


2. Eu, Fu and Yeh gave a strengthening of the Chung-Feller Theorem and a weighted version for schroder paths.

Eu, S. P. Fu, T. S. and Yeh, Y. N. Refined Chung-Feller theorems for lattice paths, J. Combin. Theory Ser. A 112 (2005) 143-162

3. Ma and Yeh gave a generalizations of Chung-Feller theorems

J. Ma, Y.N. Yeh, Generalizations of Chung-Feller theorems, Bull. Inst. Math., Acad. Sin.(N.S.)4(2009) 299-332.

第 17页

Our main results

4. Ma and Yeh gave a characterization for uniform partitions of cyclic permutations of a sequence of real number

J. Ma, Y.N. Yeh, Cyclic permutations ofsequences and uniform partitions, The electronic journal ofcombinatorics 17 (2010), #R117.

5. Liu, Wang, Yeh gave the concepts of functions of Chung-Feller type

S.C. Liu, Y. Wang, Y.N. Yeh, Chung-Feller Property in View of Generating Functions, Electron. J. Comb. 18(2011), #P104.

6. Ma and Yeh gave a refinement of Chung-Feller theorems

J. Ma, Y.N. Yeh, Refinements of (n,m)-Dyck paths, European. J. Combin. 32(2011) 92-99.

第 18页

Our main results7. Ma and Yeh generalized the cycle lemma.

J. Ma, Y.N. Yeh, Generalizations of the cycle lemma, (Accepted 2014).8. Ma and Yeh gave a characterization for uniform partitions of cyclic

permutations of a sequence of real number

J. Ma, Y.N. Yeh, Rooted cyclic permutations of a lattice paths and uniform partitions, submitted.

9. Ma and Yeh studied a class of generating functions and their functions of Chung-Feller typeJ.Ma, Y.N.Yeh, Combinatorial interpretations for a class of functions of Chung-Feller theorem. submitted

19

Part II.

Combinatorial interpretations

for a class of

function equations

20

Uniform-partition Extension

),(

21

Liu, S. C. Wang, Y. and Yeh, Y. N. The function of uniform-partition type, submitted

the function of uniform-partition type for :

.)()(

1

)()(),( Then

.1

)(

1

1

),(CS

function generating heconsider t We

.0any for Suppose .)(Let

0

001

0 0 000 0,

,0

zyz

zzSyzyzS

y

zSyzySzyCS

y

yzsyzsz

y

ys

zyszyszyfzy

nksfzszS

n

n

nn

n

nn

nn

n

n n

n

k

nkn

n

k

nkn

n

n

k

nkkn

nknn

nn

22

An example for catalan sequence (up-down type)

23

An example for Motzkin sequence (left-right type)

221 MzzMM 22 ][zMzMzzzM 21 AA

Az

2**

*

][1 AA

Aw

rightmost lowest point

第 24页

• In general, given a combinatorial structure, let f(z) be a generating function correspoding with this combinatorial structure. We can obtain a functional equation which f(z) satisfies.

第 25页

Combinatorial structure Generating

function

f(z)

Functional equation which

f(z) satisfies

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

C(z)

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

M(z)

Schroder path:(1,1),(1,-1),(2,0) in the first quadrant

S(z)

第 26页


function

f(z)


f(z) satisfies


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


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


S(z) S(z)=1+zS(z)+z[S(z)]2

第 27页


function

f(z)


f(z) satisfies


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


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


S(z) S(z)=1+zS(z)+z[S(z)]2

Given a functional equation , how to find a combinatorial structure and its corresponding generating function f(z) such that f(z) satisfies this functional equation?

第 28页


function

f(z)


f(z) satisfies


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


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


S(z) S(z)=1+zS(z)+z[S(z)]2


第 29页


function

f(z)


f(z) satisfies


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


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


S(z) S(z)=1+zS(z)+z[S(z)]2

??? ??f(z)


第 30页

Let . The recurrence relation which the sequence satisfies is independent on a0(z) . Hence, let a0(z) =1..

We focus on the following functional equation.

Let S be a set of vector in the plane Z×Z. We also call the set S step set and vectors in S steps.

Let L be a function from S to N, where N is the set of nonnegative integers . We call L a step-length function of the set S and L(s) the step length of the step s in the set S repectively.

Let W be a function from S to R, where R is the set of real numbers. We call W a weight function of the set S and W(s) the weight of the setp s in the set S respectively,

第 31页

Let P be a sequence of vectors (x1,y1)…(xn,yn) in the set S such that y1+…yn=0, y1+…yi≥0 for all i. We call P an S-path. Let Ω(S) be the set of all S-paths.

Define the L-length of a S-path P= (x1,y1)…(xn,yn) , denoted by L(P), as L(P)=L(x1,y1)+…L(xn,yn).

Define the W-length of a S-path P= (x1,y1)…(xn,yn) , denoted by W(P), as W(P)=W(x1,y1)…W(xn,yn).

Define a generating function f(z) as

第 32页

.1

equation functional thesatisfies

function generating Then the

. (n,0) to(0,0) from

paths-S of weightsof sum thebe and 1Let

.1110

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

:Theorem

1 1

0

0

ijr

i

m

jij

n

nn

n

ij

[f(z)]zaf(z)

zff(z)

ff

a),w(j,-i),w(

mjriijS

第 33页

• A decomposition of a S-path.

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

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

第 34页

Part III. Lattice paths

and Fluctuation theory

35

• Fluctuation theory is the name given to that part of probability theory which deals with the fluctuations of the partial sums sn=x1+...+xn of a sequence of random variables x1,…,xn.

36

• Consider x=(r1,…rn). Let s0=0,si=r1+…+ri

• Let p(x) be the number of positive sums si

• Let m(x) be the index where the maximum is attained for the first time.

37

x partial sum p(x) m(x)

(1,2,3) (3,1,-2) (3,4,2) 3 2

(1,3,2) (3,-2,1) (3,1,2) 3 1

(2,1,3) (1,3,-2) (1,4,2) 3 2

(3,1,2) (-2,3,1) (-2,1,2) 2 3

(2,3,1) (1,-2,3) (1,-1,2) 2 3

(3,2,1) (-2,1,3) (-2,-1,2) 1 3

r1=3,r2=1,r3=-2

38

x partial sum p(x) m(x)

(1,2,3) (1,2,-2) (1,3,1) 3 2

(1,3,2) (1,-2,2) (1,-1,1) 2 1

(2,1,3) (2,1,-2) (2,3,1) 3 2

(3,1,2) (-2,1,2) (-2,-1,1) 1 3

(2,3,1) (2,-2,1) (2,0,1) 2 1

(3,2,1) (-2,2,1) (-2,0,1) 1 3

r1=1,r2=2,r3=-2

39

• Fix X=(r1,…rn).

• Let Xi=(ri,…rn,r1,…,ri-1) (cyclic permutations.)

• Let P(X)={p(Xi)| i=1,2,…,n}

M(X)={m(Xi)| i=1,2,…,n}

40

• F. Spitzer, (1956)

• Let X be a sequence of real numbers of length n such that sn=0 and no other partial sum of distinct elements vanishes. Then P(X)=M(X)=[0,n-1].

41

Remark

• Fix X=(r1,…rn). Suppose r1+…+rn=m.• Let m=0.

The conditions in the results of Spitzer are necessary and sufficient conditions for P(X)=[0,n-1]

The conditions in the results of Spitzer are not necessary for M(X)=[0,n-1].

42

• T.V. Narayana, (1967)

• Let n be a positive integer and X be a sequence of integers with -n<ri< 2 for all i=1,2,…,n such that sn=1. Then P(X)=[n].

43

• J. Ma, Y.N. Yeh, Generalizations of The Chung-Feller Theorem II, submitted.

• Let n be a positive integer and X be a sequence of integers with -n<ri< 2 for all i=1,2,…,n such that sn=1. Then M(X)=[n].

44

Two natural problems

• What are necessary and sufficient conditions for M(X)=[n] and P(X)=[n] if m>0?

• What are necessary and sufficient conditions for M(X)=[0,n-1] and P(X)=[0,n-1] if m<=0?

45

• Fix X=(r1,…rn). Given an index j=1,…,n, define

LP(X;j)={i|sj>si,i=1,…,j-1} and

RP(X;j)={i|sj>=si i=j+1,…,n}

46

• Let m>0.The necessary and sufficient conditions for M(X)=[n] are sm(X)-si>=m for all i in LP(X;m(X))The necessary and sufficient conditions for P(X)=[n] are sj-si>=m for any j in [n] and any all i in [0,j-1]\LP(X;j)

47

• Let m<=0.The necessary and sufficient conditions for M(X)=[0,n-1] are si -sm(X)<m for all i in RP(X;m(X))

The necessary and sufficient conditions for P(X)=[0,n-1] are sj-si<m for any j in [n] and any all i in [0,j-1]\LP(X;j)

48

Part IV. Paths with some avoiding

sets shift equivalence

49

Let M be a Motzkin path.

LM: the set of the height of the level stepsLM={0,3}

PM: the set of the height of the peaks PM={2,1}

VM: the set of the height of the valleys VM={0,1}

50

Motzkin paths from (0,0) to (2(n-1),0) without level of height larger than 0

51

Peaks-, Valleys- and Level-avoiding Sets CPBA ,,

CVBPAL MMM ,,

we consider the Motzkin path such that Given the sets

A: level-avoiding set B: peak-avoiding sest C: valley-avoiding set

CVBPAL MMM ,,

A: level-restricting set B: peak-restricting setC: valley-restricting set

(1):

(2):

52

Generating FunctionsCBAslkn

m,,;,,,

the number of the Motzkin path of length n withk levles, l peaks and s valleys

slknCBAslknCBA

qyxzmqzyxM,,;,,,,,

),,,(

53

)}0(])0(1[

)0(1}{)1(1{

)0(1

,,,,

,,1,1,12

,,

,,

CqxzMAM

xzMAyBMz

xzMA

M

CBACBA

CBACBA

CBA

CBA

54

Some results

,}1{,;Pnm

,}{,; hPnm

E. Deutsch, Dyck path enumeration, Discrete Math. 204 (1999), 167--202.

P. Peart and W-J. Woan, Dyck paths with no peaks at height k,J. Integer Seq. 4 (2001), Article 01.1.3.

,}2{,;Pnm

the n-th Fine number

the (n-1)-th Catalan number

55

Shu-Chung Liu, Jun Ma, Yeong-Nan Yeh, Dyck Paths with Peak- and Valley-Avoiding Sets, Stud. Appl. Math. 121:263-289

S.-P. Eu, S.-C. Liu, and Y.-N. Yeh, Dyck Paths with Peaks Avoiding or Restricted to a Given Set, Stud. Appl. Math. 111 Iss 4 (2003), 453--465.

CBPknm

,,;,

,,;, BPknm

• Continued fractions• Close forms• Shift equivalence

56

0

)(n

nn zazH

0

)(n

nn zbzG

nsn ba

)()( zGzzH m

Suppose and

Then

if and only if there is a positive integer m such that

is a polynomial.

nsn ba

nqnp ba

If there exist non-negative integers p and q such that

57

Some interesting shift equivalence

numberCatalanmm

mmm

ssNnsNns

sNnsNnsNn

...

...

,}3,2{,;,}2{,;

}1,0{,,;}0{,,;,,;

numberCatalandGeneralizemm ssENnsNn ...

,O,;O,E,;

numberCatalandGeneralizem

mmm

mmm

sNns

sNnsNnsNns

sNnsNnsNn

...

...

...

33

333333

333

{1},}1{,;

{0},}0{,;{2},}2{,;{1},}1\{}1{,;

{0,2},,;{0,1},,;{1,2},,;

58

Some interesting shift equivalence

SloaneinAmm ssNnsNn 025265...

3333 }0{,}1{,;}2{,}0{,;

SloaneinAmm ssNnsNn 127389...

3333 }0{,}2{,;}2{,}1{,;

SloaneinAmmm ssENnsENnsNn 126120...

1,}2{)1E(,;,E,;O,O,;

numberMotzkinmm

mm

mmm

ssNnsNns

sNnsNns

sNnsNnsNn

...

...

...

,E,;,\{1}O,;

1E,}2{,;E,,;

2O,}3,2{,;1O,}2{,;O,,;

59

Continued fractionsIt is difficult to represent

as a continued fractions

),,,(,,

qzyxMCBA

60

Close form• Matrix methods

We just consider

)1,,,(),,(,,,

zyxMzyxFBABA

61

),,(]})1(1[

)0({1),,(

,1,1,12

,

zyxFyBMz

xzAzyxF

BACBA

BA

62

||and ||

|| and ||

|| and ||

|| and ||

BA

BA

BA

BA

),,(

),,(

),,(

),,(

,

,

,

,

zyxF

zyxF

zyxF

zyxF

P

N

P

),,(,

zyxFBA

63

)()1(1 22,, BiyzAixzziBA

|| and || BA

xz

Tx 2

10

1

01

xSx

}max,max{max BAm )max(max BA

Let

),,(

),,(),,(

,2

,2

, zyxFzdc

zyxFzbazyxF

mm

mm

BA

mBAiBA

STdc

ba m

imm

mm

,,,,

1

1

Then

where

64

)1(2 m

mm

mm zdc

ba

65

(A,B) being Congruence classes

} somefor )(mod|{: IjkjnnIk

),(),( kk JIBA

),,(

),,(),,(

,2

,2

, zyxFdzc

zyxFbzazyxF

kk

kk

kk

JI

JI

JI

kJIiJIST

dc

ba k

i,,,,

1

1

Define the congruence classes

Let

Then

where

66

then we say that F(x) is algebraic

0)(...)()( 01

1 xayxayxa d

dd

d

If F(x) is a solution of an equation

)(FadThe algebraic degree of F(x):

67

kk JBIA

BA

BA

BA

BA

and

||and ||

|| and ||

|| and ||

|| and ||

),,(,

zyxFBA

),,(,

zyxFBA

is algebraic since it is a solution of aquadratic Equations

2)),,((,

zyxFadBA

68

Problem I• Characterize the set

dzyxFadBABA

)),,((|),(,

69

Problem II• Given a sequence a1,a2,…,an,…, find a

pair (A,B) of the sets such that

nBAn azFz )(][

,

70

1,1,,|),,()( yxBABA

zyxFzF

BAnm

,;We consider the coefficients

in

71

|| and || ii BA

)(

)()(

,2

,2

, zFzdc

zFzbazF

ii

ii

BA ii

2211 ,;,; BAnsBAnmm

))1((

))1((22

11121

2

22222

22

2

2

1

zdzdccz

zdzdcczz r

rk

][| 21212

22121

22121 zdbcazzdbcazddcc k

)]1([

)1()1(

121212

212121

211221zdbcbdaz

zdbcbdazdddcdc

k

1)max(max}max,max{max iiiii ABBAr Let

Suppose

Thenif and only if

(1)

(2)

(3)

k Such that

72

Bijection methods

B1

}2{}1{,}0{)1(;2,; BAnBAn

mm

Suppose

Then

In fact, if B1

1)()(,

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

zFzzFBABA

73

Problem III

),(),( 2211 BABA s

2211 ,;,; BAnsBAnmm If the sequences

then we say that (A1,B1) and (A2,B2) shift equivalent,

denoted by

• Give a characterization of ),(),( 2211 BABA s

74

}1{},{; inm

is shift equivalent to the Fibonacci numbers

The sequence mn+2i;{i},{1} has Chung-Feller property, i.e., mn+2i;{i},{1} =Fn is independent on i, where Fn is the n-th Fibonacci number.

75

]1,2[},{; iinm

is shift equivalent with the Central binomial coefficients

Replace valleys(DU) of height 0 and level into peak DU and U respectively

Remove the first and final steps

(i=1)

Left factor of Dyck path

第 76页

Addition formulas and Hankel matrix

studied.y been widel has

,

sequence, theof

matrices Hankel theof tsdeterminan theevaluate

toproblem the,0 and }{ sequence aFor

1,0

0

njikji

nn

a

ka

第 77页

101,0

1,01

1,0

2det

1det

1det

,2

1

1 numbersCatalan For the

mjinjimji

njiji

njiji

n

ji

jinc

c

c

n

n

nc

第 78页

)6 (mod 4 ,3 if1

)6 (mod 5 ,2 if0

)6 (mod 1 ,0 if1

det

1det

n,length of pathsMotzkin ofnumber

count the numbersMotzkin For the

1,01

1,0

n

n

n

m

m

m

njiji

njiji

n

第 79页

.2det

2det

paths,der oSchr large ofnumber

count the numbersder oSchr large For the

2

1

1,01

2

1,0

0

n

njiji

n

njiji

nn

r

r

r

identity. Jacobi-Desnanot

by det and det

by evaluated becan det that Notice

10110

10

ni,jjini,jji

ni,jmji

aa

a

第 80页

resultsOur

.)(

by)0( offunction generating theDefine

)).0(( Denote

.))1 1(( ,))0 1(( ,1))1 1((Let

axis.- thebelownever

are that )}0 1()1 1()1 1{(set step with )( to)00(

from paths lattice weightedofset the)denote0(Let

n

rk

rkn,kn,r

n

rin

nn,k

n

xaxa

,iP

,kPwa

v,wu,w,w

x

,,-,,,Sn,k,

,kP

第 81页

Theorem

bijection. aby difference with pathssuch count

Then steps. last theofheight lowest theand step

th theofheight end ebetween th difference the

toaccording k)(0, of paths heclassify t we

, k))(0,()(][

Since :proof theofSketch

0

r

n

mr

P

Pwxaxa

nm

nmn,mk

n,km

.)()()(

,0For ),min(

00

nm

rn,rm,r

rn,m xaxavxa

m,n

第 82页

第 83页

).()(514253021)(

have we,051423324150for Then

1514253021

1449491249129

1335354

1222

11

1

)(

1514253021

149129

1354

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1

1Let Example.

,

),min(

0,

54320,5

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232

2

xaxaxxxxxxa

)},),(,),(,),(,),(,),(,{((m,n)

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xa

a

,vu

rn

nm

rrm

i,j

i,j

第 84页

Corollary .))(det( 2

)1(

1,00,

nn

njiji vxa

.)(det then ,1)( since

.

)()()()(

)()()(

)()(

)(

where

,),,,,1()()()(

:Proof

2

)1(22

)1(

1,00,,,

1,12,11,10,1

2,21,20,2

1,10,1

0,0

12

1,0

),min(

00,0,1,00,

nnnn

njijiiiii

nnnnn

Tn

nji

ji

rji

r

njiji

vAvxaaxa

xaxaxaxa

xaxaxa

xaxa

xa

A

AAvvvdiagxaxavxa

.det have wecorollary, in the 0Let :Remark 2

)1(

1-nji,0ji

nn

vax

第 85页

cases Special

,1mdet ,1)(mdet

have we,number motzkin the,1 and 1 case For the

.1det

then,number catalan that thenotice ,0let

1(x)det

have we theoremby the ,paritydifferent have and if 0 Since

.)( Denote axis.- below

passnever that (1,-1) and (1,1) steps using )22( to)00( frompath of

number theis where, have we,1 0 case For the

1,0ji1,0j,0i

0

1,0

020

1

1,00,

22,

njinji

n,n

njiji

n,n,n

n

njiji

n,k

n

ri

rin,in,r

n,kkn,kn

x

amvu

c

accx

xc

kna

xcxcx

kn,,

cacv,u

第 86页

. )( and similar to defined are )( and r

. and 1 are steps of weights the,)0,2(),1,1(,1,1Let

,kn,,kn, xaaxr

u,vS

rnrn

pathsder oSchr Large

Theorem

.)()()()()(

,1For ),min(

0

)1,1min(

0110

nm

rn,rm,r

rnm

r,rn,rm

rn,m xaxavxaxavuxr

m,n

第 87页

第 88页

Thank you for your attention!

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