A note on the distribution of major index for Schr\"oder paths

Xiaomei Chen

arXiv:1906.09018·math.CO·June 24, 2019

A note on the distribution of major index for Schr\"oder paths

Xiaomei Chen

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TL;DR

This paper corrects an error in previous formulas regarding the distribution of the major index for Schr"oder paths and provides a complete proof for all cases, enhancing the mathematical understanding of these combinatorial objects.

Contribution

It rectifies a mistake in earlier work and extends the proof to cover all cases of the distribution of the major index for Schr"oder paths.

Findings

01

Corrected the proof of the distribution formulas for Schr"oder paths

02

Extended the proof to all cases beyond E<D<N

03

Ensured the formulas are valid universally

Abstract

Bonin, Shapiro and Simion (1993) gave two formulas on the distribution of major index for Schr\"oder paths, and proved their result for the case $E < D < N$ . In this short note, we correct an error in their proof, and give a complete proof for all cases.

Equations32

M S c h_{L} (n, l; q) :

M S c h_{L} (n, l; q) :

= \frac{1}{[ l - n + 1 ]} [l - n 2 ( l - n )] [2 n - l l], i f E < N,

M S c h_{L} (n, l; q) :

M S c h_{L} (n, l; q) :

= \frac{q ^{l - n}}{[ l - n + 1 ]} [l - n 2 ( l - n )] [2 n - l l], i f E > N .

M D e l (m, n, l; q) := W \in D e l (m, n, l) \sum q^{maj (W)}

M D e l (m, n, l; q) := W \in D e l (m, n, l) \sum q^{maj (W)}

M B D e l (m, n, l; q) := W \in B D e l (m, n, l) \sum q^{maj (W)}

M B D e l (m, n, l; q) := W \in B D e l (m, n, l) \sum q^{maj (W)}

M B D e l (n, n, l; q) = q M D e l (n + 1, n - 1, l; q), i f E < N,

M B D e l (n, n, l; q) = q M D e l (n + 1, n - 1, l; q), i f E < N,

M B D e l (n, n, l; q) = M D e l (n + 1, n - 1, l; q), i f E > N .

M B D e l (n, n, l; q) = M D e l (n + 1, n - 1, l; q), i f E > N .

φ : B D e l (n, n, l) \to D e l (n + 1, n - 1, l)

φ : B D e l (n, n, l) \to D e l (n + 1, n - 1, l)

W_{1} := w_{k - r} w_{k - r + 1} \dots w_{k - 1} w_{k} w_{k + 1} \dots w_{k + s}

W_{1} := w_{k - r} w_{k - r + 1} \dots w_{k - 1} w_{k} w_{k + 1} \dots w_{k + s}

\mathrm{maj}(W)=\left\{\begin{array}[]{ll}\mathrm{maj}(\varphi(w))+1,&\hbox{if $E<N$;}\\ \mathrm{maj}(\varphi(w)),&\hbox{if $E>N$.}\end{array}\right.

\mathrm{maj}(W)=\left\{\begin{array}[]{ll}\mathrm{maj}(\varphi(w))+1,&\hbox{if $E<N$;}\\ \mathrm{maj}(\varphi(w)),&\hbox{if $E>N$.}\end{array}\right.

W_{1} := w_{k - r} w_{k - r + 1} \dots w_{k - 1} w_{k} w_{k + 1} \dots w_{k + s}

W_{1} := w_{k - r} w_{k - r + 1} \dots w_{k - 1} w_{k} w_{k + 1} \dots w_{k + s}

M D e l (m, n, l; q) = [l - m , l - n , m + n - l l] .

M D e l (m, n, l; q) = [l - m , l - n , m + n - l l] .

M S c h_{L} (n, l; q) = M D e l (n, n, l; q) - M B D e l (n, n, l; q) .

M S c h_{L} (n, l; q) = M D e l (n, n, l; q) - M B D e l (n, n, l; q) .

M S c h_{L} (n, l; q)

M S c h_{L} (n, l; q)

= \frac{1}{[ l - n + 1 ]} [l - n 2 ( l - n )] [2 n - l l],

M S c h_{L} (n, l; q)

M S c h_{L} (n, l; q)

= \frac{q ^{l - n}}{[ l - n + 1 ]} [l - n 2 ( l - n )] [2 n - l l] .

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TopicsAdvanced Combinatorial Mathematics · Random Matrices and Applications · Mathematical Dynamics and Fractals

Full text

A note on the distribution of major index for Schröder paths

Xiaomei Chen

Abstract

Bonin, Shapiro and Simion (1993) gave two formulas on the distribution of major index for Schröder paths, and proved their result for the case $E<D<N$ . In this short note, we correct an error in their proof, and give a complete proof for all cases.

1 Introduction

For the notation and terminology below on lattice paths, see [1]. Let $Del(m,n,l)$ denote the set of Delannoy paths from $(0,0)$ to $(m,n)$ with $l$ steps, where a Delannoy path is a lattice path using only the three steps (1,0), (1,1) and (0,1). A Schröder $n-$ path is a Delannoy path from $(0,0)$ to $(n,n)$ which never goes above the diagonal line $y=x$ , and we use $Sch_{L}(n,l)$ to denote the collection of all Schröder $n-$ paths with $l$ steps.

In the following, we use $E$ , $D$ and $N$ to denote the three steps (1,0), (1,1) and (0,1) respectively, and represent a Delannoy path of length $l$ by a word $W=w_{1}w_{2}\cdots w_{l}$ over the alphabet set $\{E,D,N\}$ . Given a linear ordering of $\{E,D,N\}$ , $i(1\leq i\leq n-1)$ is called a descent of $W$ if $w_{i}>w_{i+1}$ . We use $D(W)$ to denote the set of all descents of $W$ , and define the major index of $W$ by $\mathrm{maj}(W):=\sum_{i\in D(W)}i$ .

Bonin et al. studied the major index for Schröder paths and gave the following result.

Theorem 1.1.

[1*]**

For given $n$ , $l$ and linear ordering of $\{E,D,N\}$ , the distribution of the statistic maj over $Sch_{L}(n,l)$ is*

[TABLE]

and

[TABLE]

Bonin et al. gave a detailed proof of the above result for the case $E<D<N$ , and omitted proof of other cases. In their proof, a path $p\in Del(n,n,l)$ is called a ’bad’ one if it runs above the line $y=x$ , and a correspondence $\psi$ from the set consisting of all bad paths in $Del(n,n,l)$ to the set $Del(n+1,n-1,l)$ is defined as follows.

For a ’bad’ path $W=w_{1}w_{2}\cdots w_{l}\in Del(n,n,l)$ , let $w_{i}$ be the first step of $W$ running above the line $y=x$ , and let $w_{j}$ be the last element of the sequence of consecutive $N$ beginning at $w_{i}$ . Then $\psi(W)$ is defined to be the path obtained from $W$ by replacing $w_{j}$ with $E$ . The correctness of the proof given in [1] relies on the statement that $\psi$ is a bijection, which however is not true. See the following example for instance.

Example 1.2.

Let $w=NENNEE$ and $w^{\prime}=EENNNE$ be two bad paths in $Del(3,3,6)$ . Then $\psi(w)=\psi(w^{\prime})=EENNEE$ .

2 Proof of Theorem 1.1

Let $BDel(m,n,l)$ denote the set of all ’bad’ paths in $Del(m,n,l)$ . Given a linear ordering of $\{E,D,N\}$ , we define

[TABLE]

and

[TABLE]

to be the distributions of the maj statistic over $Del(m,n,l)$ and $BDel(m,n,l)$ respectively.

For a path $W=w_{1}w_{2}\cdots w_{l}\in Del(m,n,l)$ , the depth of $w_{i}(1\leq i\leq l)$ is defined to be the difference between the number of $N$ and the number of $E$ in the subpath $w_{1}w_{2}\cdots w_{i}$ . By extending the technique applied to Catalan paths in [2], we obtain the following result.

Lemma 2.1.

For given $n,l$ and linear ordering of $\{E,D,N\}$ , we have

[TABLE]

and

[TABLE]

Proof.

We prove Lemma 2.1 by constructing a bijection

[TABLE]

as follows. Given a path $W=w_{1}w_{2}\cdots w_{l}\in BDel(n,n,l)$ , let $w_{k}$ be its first deepest step. We denote by

[TABLE]

the subpath of $W$ such that $w_{k-r-1}\neq D$ , $w_{k+s+1}\neq D$ , and $w_{k-i}=w_{k+j}=D$ for $1\leq i\leq r$ and $1\leq j\leq s$ . Then $\varphi(W)$ is defined as follows.

(1)

If $E<D<N$ , $N<E<D$ or $D<N<E$ , $\varphi(W)$ is obtained from $W$ by replacing $w_{k}$ with $E$ . 2. (2)

If $E<N<D$ , $D<E<N$ or $N<D<E$ , $\varphi(W)$ is obtained from $W$ according to the following two cases:

•

when $r\geq 1$ , replacing $W_{1}$ with the path $\underbrace{DD\cdots D}_{r-1}E\underbrace{DD\cdots D}_{s+1}$ ;

•

when $r=0$ , replacing $W_{1}$ with the path $\underbrace{DD\cdots D}_{s}E$ .

By the definition of $w_{k}$ and $W_{1}$ , we must have $w_{k}=N$ , $w_{k+s+1}=E$ , and $w_{k-r-1}=N$ if $k-r-1\geq 1$ . Then it is not difficult to verify that

[TABLE]

For instance, if $E<N<D$ and $r=0$ , then $D(\varphi(W))=D(W)-\{k+s\}\cup\{k-s-1\}$ , which implies that $\mathrm{maj}(W)=\mathrm{maj}(\varphi(W))+1$ .

Thus to complete the proof of Lemma 2.1, it is enough to show that $\varphi$ is a bijection. Given a path $W=w_{1}w_{2}\cdots w_{l}\in Del(n+1,n-1,l)$ , let $w_{k-1}$ be the last of its deepest step, where the depth of $w_{0}$ is defined to be 0. Then we must have $w_{k}=E$ . We denote by

[TABLE]

the subpath of $W$ such that $w_{k-r-1}\neq D$ , $w_{k+s+1}\neq D$ , and $w_{k-i}=w_{k+j}=D$ for $1\leq i\leq r$ and $1\leq j\leq s$ . Then the path $\varphi^{-1}(W)$ is constructed as follows.

(1)

If $E<D<N$ , $N<E<D$ or $D<N<E$ , $\varphi^{-1}(W)$ is obtained from $W$ by replacing $w_{k}$ with $N$ . 2. (2)

If $E<N<D$ , $D<E<N$ or $N<D<E$ , $\varphi^{-1}(W)$ is obtained from $W$ according to the following two cases:

•

when $s\geq 1$ , replacing $W_{1}$ with the path $\underbrace{DD\cdots D}_{r+1}N\underbrace{DD\cdots D}_{s-1}$ ;

•

when $s=0$ , replacing $W_{1}$ with the path $N\underbrace{DD\cdots D}_{r}$ .

It is obvious that the above construction gives the inverse of $\varphi$ , thus we completes the proof. ∎

We are now ready to give the proof of Theorem 1.1.

Proof.

By a result of MacMahon([3]) on the distribution of maj over all permutations of a multiset, we have

[TABLE]

It is obvious that

[TABLE]

Therefore by Lemma 2.1, for the case when $E<N$ , we have

[TABLE]

and for the case when $E>N$ , we have

[TABLE]

∎

Bibliography3

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] Bonin J, Shapiro L, Simion R. Some q-analogues of the Schröder numbers arising from combinatorial statistics on lattice paths[J]. Journal of Statistical Planning and Inference, 1993, 34(1): 35-55.
2[2] Fürlinger J, Hofbauer J. q-Catalan numbers[J]. Journal of Combinatorial Theory, Series A, 1985, 40(2): 248-264.
3[3] Mac Mahon P A. Combinatory analysis(Vol. 2)[M]. Cambridge University Press, Cambridge, 1918. Reprinted by Chelsea, New York, 1960.

TL;DR

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Abstract

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A note on the distribution of major index for Schröder paths

Abstract

1 Introduction

Theorem 1.1**.**

Example 1.2**.**

2 Proof of Theorem 1.1

Lemma 2.1**.**

Proof.

Proof.

Theorem 1.1.

Example 1.2.

Lemma 2.1.