Proof of Chapoton's conjecture on Newton polygons of $q$-Ehrhart polynomials

TL;DR

This paper proves Chapoton's conjecture regarding the shape of the Newton polygon of the numerator of the $q$-Ehrhart polynomial for order polytopes, advancing understanding of $q$-analog combinatorics.

Contribution

The paper provides a rigorous proof of Chapoton's conjecture on the Newton polygon shape for $q$-Ehrhart polynomials of order polytopes, a previously unconfirmed geometric property.

Findings

Confirmed the shape of the Newton polygon as conjectured by Chapoton.

Established new properties of $q$-Ehrhart polynomials related to their Newton polygons.

Enhanced understanding of the geometric structure of $q$-analog polynomials.

Abstract

Recently, Chapoton found a $q$-analog of Ehrhart polynomials, which are polynomials in $x$ whose coefficients are rational functions in $q$. Chapoton conjectured the shape of the Newton polygon of the numerator of the $q$-Ehrhart polynomial of an order polytope. In this paper, we prove Chapoton's conjecture.

Equations

$$nM=\{n\mathbf{x}:\mathbf{x}\in M\}.$$

$$W(M,q)=\sum _{\mathbf{x}\in M\cap {\mathbb{Z}}^m}{q}^{|\mathbf{x}|},$$

$$|\mathbf{x}|=x_1+\cdots +x_m.$$

$$[n{]}_q:=\frac{1-q^n}{1-q},$$

$$[n{]}_q!:=[1{]}_q[2{]}_q\dots [n{]}_q,{\left[\genfrac{}{}{0ex}{}{n}{k}\right]}_q:=\frac{[n{]}_q!}{[k{]}_q![n-k{]}_q!}.$$

$$[n{]}_q$$

$$[-n{]}_q$$

$$[a+b{]}_q$$

$$E([n{]}_q)=W(nM,q).$$

$$E([-n{]}_q)=(-1{)}^{\dim M}W(n{M}^{\circ },1/q).$$

$$\mathcal{O}(P)=\{(x_1,\dots ,x_m)\in [0,1{]}^m:x_i\le x_j\text{mbox}ifi{\le }_{P}j\}.$$

$$E_P(x)=\frac{N_P(q,x)}{D(q)},$$

$$\mathrm{Newton}(N_P(q,x))=C(a_1,\dots ,a_m;h),$$

$$\sum _{\sigma \in \mathcal{P}(P,n)}{q}^{|\sigma |}=\sum _{\pi \in \mathcal{L}(P)}{q}^{\mathrm{maj}(\pi )}{\left[\genfrac{}{}{0ex}{}{n-\mathrm{des}(\pi )+m}{m}\right]}_q.$$

$$\mathcal{P}(P)=\underset{\pi \in \mathcal{L}(P)}{⨄}{S}_{\pi }.$$

$$\mathcal{P}(P,n)=\underset{\pi \in \mathcal{L}(P)}{⨄}{S}_{\pi }(n).$$

$$\sum _{\sigma \in \mathcal{P}(P,n)}{q}^{|\sigma |}=\sum _{\pi \in \mathcal{L}(P)}\sum _{\sigma \in S_{\pi }(n)}{q}^{|\sigma |}=\sum _{\pi \in \mathcal{L}(P)}\sum _{\begin{array}{c}n\ge i_1\ge \cdots \ge i_m\ge 0\\ i_j>i_{j+1}\mathrm{if}j\in \mathrm{Des}(\pi )\end{array}}{q}^{i_1+\cdots +i_m}.$$

$$\sum _{\begin{array}{c}n\ge i_1\ge \cdots \ge i_m\ge 0\\ i_j>i_{j+1}\mathrm{if}j\in \mathrm{Des}(\pi )\end{array}}{q}^{i_1+\cdots +i_m}=q^{\mathrm{maj}(\pi )}{\left[\genfrac{}{}{0ex}{}{n-\mathrm{des}(\pi )+m}{m}\right]}_q,$$

$$W(n\mathcal{O}(P),q)=\sum _{\sigma \in \mathcal{P}(\overline{P},n)}{q}^{|\sigma |}.$$

$$E_P(x)=\frac{1}{[m{]}_q!}\sum _{\pi \in \mathcal{L}(\overline{P})}{q}^{\mathrm{maj}(\pi )}\prod _{i=1}^m([i-\mathrm{des}(\pi ){]}_q+q^{i-\mathrm{des}(\pi )}x).$$

$$f([n{]}_q)=\sum _{\pi \in \mathcal{L}(\overline{P})}{q}^{\mathrm{maj}(\pi )}\frac{{\prod }_{i=1}^m[i-\mathrm{des}(\pi )+n{]}_q}{[m{]}_q!}=\sum _{\pi \in \mathcal{L}(\overline{P})}{q}^{\mathrm{maj}(\pi )}{\left[\genfrac{}{}{0ex}{}{n-\mathrm{des}(\pi )+m}{m}\right]}_q.$$

$$W(n\mathcal{O}(P),q)=\sum _{\pi \in \mathcal{L}(\overline{P})}{q}^{\mathrm{maj}(\pi )}{\left[\genfrac{}{}{0ex}{}{n-\mathrm{des}(\pi )+m}{m}\right]}_q.$$

$$F_P(q,x)=\sum _{\pi \in \mathcal{L}(P)}{q}^{\mathrm{maj}(\pi )}\prod _{i=1}^m([i-\mathrm{des}(\pi ){]}_q+q^{i-\mathrm{des}(\pi )}x).$$

$$\mathrm{maj}(\pi )+\sum _{i=1}^{\mathrm{des}(\pi )}(i-\mathrm{des}(\pi ))\ge \left(\genfrac{}{}{0ex}{}{\mathrm{des}(\pi )+1}{2}\right)+\left(\genfrac{}{}{0ex}{}{\mathrm{des}(\pi )+1}{2}\right)-\mathrm{des}(\pi {)}^2\ge 0.$$

$$F_P(q,x)=[m{]}_q!E_{\overline{P}}(x).$$

$$\mathrm{Newton}(N_P(q,x))=C(a_1,\dots ,a_m;h),$$

$$\mathrm{Newton}(F_{\overline{P}}(q,x))=C(a_1,\dots ,a_m;h^{\prime }),$$

$$F_{\overline{P}}(q,x)=N_P(q,x)\varphi (q).$$

$$\varphi (q)=q^r+c_{r-1}{q}^{r-1}+\cdots +c_1{q}^1+1,$$

$$\max \{i:[q^i{x}^k]N_P(q,x)\ne 0\}=\max \{i:[q^{i+r}{x}^k]F_{\overline{P}}(q,x)\ne 0\},$$

$$\min \{i:[q^i{x}^k]N_P(q,x)\ne 0\}=\min \{i:[q^i{x}^k]F_{\overline{P}}(q,x)\ne 0\},$$

Full text

This work was supported by NRF grants #2016R1D1A1A09917506 and #2016R1A5A1008055.

1. Introduction

In 1962, Ehrhart [5] discovered certain polynomials associated to lattice polytopes. These polynomials are now widely known and called Ehrhart polynomials. They contain important information of lattice polytopes such as the number of lattice points in the polytope, the number of lattice points in the relative interior and the relative volume of the polytope.

Recently, Chapoton [4] found a $q$-analog of Ehrhart polynomials and generalized some properties of them. A $q$-Ehrhart polynomial is a polynomial in variable $x$ whose coefficients are rational functions in $q$. Thus we can write a $q$-Ehrhart polynomial as a rational function in $q$ and $x$ whose numerator is a polynomial in $q$ and $x$, and whose denominator is a polynomial in $q$. In the same paper, Chapoton conjectured the shape of the Newton polygon of the numerator of the $q$-Ehrhart polynomial associated to an order polytope. The goal of this paper is to prove Chapoton’s conjecture.

First, we briefly review basic properties of Ehrhart polynomials and their $q$-analogs. See [1, 3, 2] for more details in Ehrhart polynomials.

A point in $\mathbb{R}^{m}$ is called a lattice point if all the coordinates are integers. A lattice polytope is a polytope whose vertices are lattice points. All polytopes considered in this paper are lattice polytopes.

For a polytope $M$ and an integer $n$, we denote by $nM$ the dilation of $M$ by a scale factor of $n$, i.e.,

[TABLE]

For a lattice polytope $M$ in $\mathbb{R}^{m}$, there exists a polynomial $E(x)$, called the Ehrhart polynomial of $M$, satisfying the following interesting properties:

• •

$E(n)=|nM\cap\mathbb{Z}^{m}|$ for all integers $n\geq 0$.

• •

$(-1)^{\dim M}E(-n)=|nM^{\circ}\cap\mathbb{Z}^{m}|$ for all integers $n\geq 0$, where $\dim M$ is the dimension of $M$ and $M^{\circ}$ is the relative interior of $M$.

• •

The degree of $E(x)$ is equal to the dimension of $M$.

• •

The leading coefficient of $E(x)$ is equal to the relative volume of $M$.

For a polytope $M$ in $\mathbb{R}^{m}$, let

[TABLE]

where for $\mathbf{x}=(x_{1},\dots,x_{m})$ we denote

[TABLE]

We use the standard notation for $q$-integers: for $n\in\mathbb{Z}$,

[TABLE]

and, for integers $n\geq k\geq 0$,

[TABLE]

Note that for $n\geq 0$ and $a,b\in\mathbb{Z}$, we have

[TABLE]

Chapoton [4, Theorem 3.1] found a $q$-analog of Ehrhart polynomials as follows.

Theorem 1.1** (Chapoton).**

Let $M$ be a polytope satisfying the following conditions:

• •

For every vertex $\mathbf{x}$ of $M$, we have $|\mathbf{x}|\geq 0$.

• •

For every edge between two vertices $\mathbf{x}$ and $\mathbf{y}$ of $M$, we have $|\mathbf{x}|\neq|\mathbf{y}|$.

Then there is a polynomial $E(x)\in\mathbb{Q}(q)[x]$ such that for every integer $n\geq 0$,

[TABLE]

The polynomial $E(x)$ in Theorem 1.1 is called the $q$-Ehrhart polynomial of the polytope $M$. We note that in [4], more generally, Chapoton considers a linear form $\lambda(\mathbf{x})$ on $\mathbb{R}^{m}$ in place of $|\mathbf{x}|$. In this setting with a linear form, Chapoton [4, Theorem 3.5] also shows a nice $q$-analog of the Ehrhart-Macdonald reciprocity:

[TABLE]

We note that Kim and Stanton [6, Theorem 9.3] showed that the leading coefficient of the $q$-Ehrhart polynomial of an order polytope is equal to the $q$-volume of the order polytope, which is defined as a Jackson’s $q$-integral over the order polytope.

In order to state Chapoton’s conjecture we need some notation and terminology.

For a polynomial $f(x_{1},\dots,x_{k})$ in $x_{1},\dots,x_{k}$, we denote by $[x_{1}^{i_{1}}\dots x_{k}^{i_{k}}]f(x_{1},\dots,x_{k})$ the coefficient of $x_{1}^{i_{1}}\dots x_{k}^{i_{k}}$ in $f(x_{1},\dots,x_{k})$. For a polynomial $f(x_{1},\dots,x_{k})$ in $x_{1},\dots,x_{k}$, the Newton polytope of $f(x_{1},\dots,x_{k})$, denoted by $\mathrm{Newton}(f(x_{1},\dots,x_{k}))$, is the convex hull of the points $(i_{1},\dots,i_{k})$ such that $[x_{1}^{i_{1}}\dots x_{k}^{i_{k}}]f(x_{1},\dots,x_{k})\neq 0$. In this paper, we consider Newton polygons, which are Newton polytopes of two-variable functions.

For a poset $P$ on $\{1,2,\dots,m\}$, the order polytope $\mathcal{O}(P)$ of $P$ is defined by

[TABLE]

As mentioned in [4], using the properties of vertices and edges of an order polytope in [7] one can check that every order polytope satisfies the conditions in Theorem 1.1. Therefore, we can consider the $q$-Ehrhart polynomial of an order polytope.

Let $E_{P}(x)$ be the $q$-Ehrhart polynomial of $\mathcal{O}(P)$. We denote by $N_{P}(q,x)$ be the numerator of $E_{P}(x)$. More precisely, $N_{P}(q,x)$ is the unique polynomial in $\mathbb{Z}[q,x]$ with positive leading coefficient such that

[TABLE]

for some polynomial $D(q)\in\mathbb{Z}[q]$ with $\gcd(N_{P}(q,x),D(q))=1$.

For integers $1\leq a_{1}\leq a_{2}\leq\dots\leq a_{m}$ and $h\geq a_{1}+\dots+a_{m}$, we define $C(a_{1},\dots,a_{m};h)$ to be the convex hull of the points $(0,0)$, $(a_{1}+\dots+a_{i},i)$ for $1\leq i\leq m$, $(h,m)$ and $(h-m,0)$. See Figure 1 for an example.

Let $P$ be a poset and $x\in P$. A chain ending at $x$ (resp. starting at $x$) is a subset $\{t_{1}<_{P}\dots<_{P}t_{k}\}$ of $P$ with $t_{k}=x$ (resp. $t_{1}=x$). The size of a chain is the number of elements in the chain. We denote by $\operatorname{mc}_{P}(x)$ the maximum size of a chain ending at $x$. We also denote by $\overline{\operatorname{mc}}_{P}(x)$ the maximum size of a chain starting at $x$. When there is no possible confusion, we will simply write as $\operatorname{mc}(x)$ and $\overline{\operatorname{mc}}(x)$ instead of $\operatorname{mc}_{P}(x)$ and $\overline{\operatorname{mc}}_{P}(x)$.

In [4, Conjecture 5.3], Chapoton proposed the following conjecture on the shape of the Newton polygon of $N_{P}(q,x)$.

Conjecture 1.2**.**

Let $P$ be a poset on $\{1,2,\dots,m\}$. Suppose that $a_{1}\leq a_{2}\leq\dots\leq a_{m}$ is the increasing rearrangement of $\overline{\operatorname{mc}}(1),\dots,\overline{\operatorname{mc}}(m)$. Then the Newton polygon of the numerator of the $q$-Ehrhart polynomial of $\mathcal{O}(P)$ is given by

[TABLE]

for some integer $h\geq a_{1}+\dots+a_{m}$.

The goal of this paper is to prove Conjecture 1.2. As Chapoton points out in [4], the $q$-Ehrhart polynomial $E_{P}(x)$ of $\mathcal{O}(P)$ can be understood as a generating function for $P$-partitions of $\overline{P}$, the dual poset of $P$. It is well-known that the generating function for $P$-partitions can be expressed in terms of linear extensions of the poset. One of the main ingredients of our proof of Conjecture 1.2 is Corollary 3.5, which gives a description of the minimum of $\operatorname{maj}(\pi)-k\operatorname{des}(\pi)$ over all linear extensions $\pi$ of $P$.

The rest of this paper is organized as follows. In Section 2, we recall necessary definitions and state our main result (Theorem 2.5), which describes the precise shape of the Newton polygon of $[m]_{q}!E_{P}(x)$. Then we show that Theorem 2.5 implies Conjecture 1.2. In Section 3 we find some property of the linear extensions of a poset. In Section 4 we prove Theorem 2.5.

2. The main result

In this section we state our main theorem, which implies Conjecture 1.2.

We first recall some definitions on permutations and posets. We refer the reader to [8] for more details.

The set of nonnegative integers is denoted by $\mathbb{N}$.

Let $\mathfrak{S}_{m}$ be the set of permutations of $\{1,2,\dots,m\}$. For $\pi=\pi_{1}\dots\pi_{m}\in\mathfrak{S}_{m}$, a descent of $\pi$ is an integer $1\leq i\leq m-1$ such that $\pi_{i}>\pi_{i+1}$. We denote by $\operatorname{Des}(\pi)$ the set of descents of $\pi$. We define $\operatorname{maj}(\pi)=\sum_{i\in\operatorname{Des}(\pi)}i$ and $\operatorname{des}(\pi)=|\operatorname{Des}(\pi)|$.

Let $P$ be a poset on $\{1,2,\dots,m\}$. A $P$-partition is an order-reversing map $\sigma:P\to\mathbb{N}$, i.e., $\sigma(x)\geq\sigma(y)$ if $x\leq_{P}y$. For a $P$-partition $\sigma$, let $|\sigma|=\sigma(1)+\dots+\sigma(m)$. We denote by $\mathcal{P}(P)$ the set of $P$-partitions. For an integer $n$, we denote by $\mathcal{P}(P,n)$ the set of $P$-partitions $\sigma$ satisfying $\sigma(x)\leq n$ for all $x\in P$.

We say that $P$ is naturally labeled if $x\leq_{P}y$ implies $x\leq y$. A linear extension of $P$ is a permutation $\pi=\pi_{1}\dots\pi_{m}\in\mathfrak{S}_{m}$ such that $\pi_{i}\leq_{P}\pi_{j}$ implies $i\leq j$. We denote by $\mathcal{L}(P)$ the set of linear extensions of $P$. Note that if $P$ is naturally labeled, $\mathcal{L}(P)$ always contains the identity permutation.

We need the following lemma, which gives a connection between certain generating functions for $\mathcal{P}(P,n)$ and $\mathcal{L}(P)$.

Lemma 2.1**.**

For a naturally labeled poset $P$ on $\{1,2,\dots,m\}$, we have

[TABLE]

Proof.

For a permutation $w\in\mathfrak{S}_{m}$, let $S_{w}$ denote the set of all functions $f:P\to\mathbb{N}$ satisfying the following conditions:

• •

$f(w_{1})\geq f(w_{2})\geq\cdots\geq f(w_{m})$ and

• •

$f(w_{i})>f(w_{i+1})$ if $i\in\operatorname{Des}(w)$.

It is well known [8, Lemma 3.15.3] that

[TABLE]

Let $S_{\pi}(n)=S_{\pi}\cap\mathcal{P}(P,n)$. Then we have

[TABLE]

Thus,

[TABLE]

It is shown in [6, Lemma 4.5] that

[TABLE]

which completes the proof. ∎

For a poset $P$, we denote its dual by $\overline{P}$, that is, $x\leq_{P}y$ if and only if $y\leq_{\overline{P}}x$. By definition, for a poset $P$ and an integer $n\in\mathbb{N}$, we have

[TABLE]

Therefore, the $q$-Ehrhart polynomial $E_{P}(x)$ of $\mathcal{O}(P)$ is closely related to $P$-partitions of $\overline{P}$. The next proposition shows that $E_{P}(x)$ can be written as a generating function for linear extensions of $\overline{P}$.

Proposition 2.2**.**

Let $P$ be a poset on $\{1,2,\dots,m\}$. Suppose that $\overline{P}$ is naturally labeled. Then the $q$-Ehrhart polynomial of $\mathcal{O}(P)$ is

[TABLE]

Proof.

Let $f(x)$ be the right hand side. Then

[TABLE]

On the other hand, by Lemma (2.1) and (1), we have

[TABLE]

Thus $f([n]_{q})=W(n\mathcal{O}(P),q)$ for all $n\in\mathbb{N}$ and we obtain $E_{P}(x)=f(x)$. ∎

Now we define a polynomial $F_{P}(q,x)$ in $q$ and $x$, which will be used throughout this paper.

Definition 2.3**.**

For a poset $P$ on $\{1,2,\dots,m\}$, we define

[TABLE]

Note that we always have $F_{P}(q,x)\in\mathbb{Z}[q,x]$ because for every $\pi\in\mathcal{L}(P)$, the power of $q$ in each summand is at least

[TABLE]

Proposition 2.2 implies that for a naturally labeled poset $P$ on $\{1,2,\dots,m\}$, we have

[TABLE]

Proposition 2.4**.**

Let $P$ be a poset on $\{1,2,\dots,m\}$ such that $\overline{P}$ is naturally labeled. Suppose that $a_{1}\leq a_{2}\leq\dots\leq a_{m}$ is the increasing rearrangement of $\overline{\operatorname{mc}}(1),\dots,\overline{\operatorname{mc}}(m)$. Then we have

[TABLE]

for some $h\geq a_{1}+\dots+a_{m}$ if and only if

[TABLE]

for some $h^{\prime}\geq a_{1}+\dots+a_{m}$. Moreover, in this case we always have $h^{\prime}=h+r$, where $r=\deg\phi(q)$ and $\phi(q)=\gcd(F_{\overline{P}}(q,x),[m]_{q}!)$.

Proof.

By (2), we have

[TABLE]

Since $\phi(q)$ divides $[m]_{q}!$, the leading coefficient and the constant term of $\phi(q)$ are both $1$. Thus, we have

[TABLE]

for some $c_{1},\dots,c_{r-1}\in\mathbb{Z}$. Hence, for each $1\leq k\leq m$, we have

[TABLE]

[TABLE]

which imply the statement. ∎

Now we state our main theorem.

Theorem 2.5**.**

Let $P$ be a naturally labeled poset on $\{1,2,\dots,m\}$. Let $b_{1}\leq b_{2}\leq\dots\leq b_{m}$ be the increasing rearrangement of $\operatorname{mc}(1),\operatorname{mc}(2),\dots,\operatorname{mc}(m)$. Then the Newton polygon of

[TABLE]

is given by

[TABLE]

We prove Theorem 2.5 in Section 4. Note that in Theorem 2.5 we have

[TABLE]

which follows from the fact that $b_{1}=1$ and $b_{i+1}\leq b_{i}+1$ for all $i$.

We finish this section by showing that Theorem 2.5 implies Conjecture 1.2.

Proof of Conjecture 1.2.

Note that relabeling of $P$ does not affect $E_{P}(q,x)$. Hence, we can assume that $\overline{P}$ is naturally labeled. Observe that $\overline{\operatorname{mc}}_{P}(x)=\operatorname{mc}_{\overline{P}}(x)$ for all $x\in\{1,2,\dots,m\}$. By Theorem 2.5,

[TABLE]

By Proposition 2.4, we obtain that

[TABLE]

for some integer $h\geq a_{1}+\dots+a_{m}$. This completes the proof. ∎

3. Some properties of linear extensions

In this section we prove some properties of posets which will be used in the next section.

Lemma 3.1**.**

Let $P$ be a naturally labeled poset on $\{1,2,\dots,m\}$ and $\pi\in\mathcal{L}(P)$. Suppose that $\operatorname{Des}(\pi)\neq\emptyset$ and $c$ is the largest descent of $\pi$. Then there is a permutation $\sigma\in\mathcal{L}(P)$ such that $\operatorname{Des}(\sigma)=\operatorname{Des}(\pi)\setminus\{c\}$.

Proof.

Let $j$ be the second largest descent of $\pi$. If $c$ is the only descent of $\pi$, we set $j=0$. Let $\sigma=\pi_{1}\dots\pi_{j}\pi_{j+1}^{\prime}\dots\pi_{m}^{\prime}$, where $\pi_{j+1}^{\prime}<\cdots<\pi_{m}^{\prime}$ is the increasing rearrangement of $\pi_{j+1},\dots,\pi_{m}$.

We claim that $\sigma\in\mathcal{L}(P)$. Consider two elements $x$ and $y$ with $x<_{P}y$. If $y$ is in $\pi_{1}\dots\pi_{j}$, since $\pi$ is a linear extension, $x$ must appear before $y$. Otherwise $y$ is in the increasing sequence $\pi_{j+1}^{\prime}\dots\pi_{m}^{\prime}$. Since $P$ is naturally labeled, $x<y$ and $x$ cannot appear after $y$ in $\pi_{j+1}^{\prime}\dots\pi_{m}^{\prime}$. Thus we always have $x$ before $y$ in $\sigma$.

Since $\pi_{j}>\pi_{j+1}\geq\pi_{j+1}^{\prime}$, we have $\operatorname{Des}(\sigma)=\operatorname{Des}(\pi)\setminus\{c\}$.

∎

Definition 3.2**.**

Let $\pi=\pi_{1}\dots\pi_{m}\in\mathfrak{S}_{m}$. A descent block of $\pi$ is a maximal consecutive subsequence of $\pi$ which is in decreasing order. We denote by $\operatorname{DB}_{i}(\pi)$ the set of elements in the $i$th descent block of $\pi$.

For example, if $\pi=384196725$, then the descent blocks of $\pi$ are $3$, $841$, $96$, $72$, $5$, and $\operatorname{DB}_{1}(\pi)=\{3\}$, $\operatorname{DB}_{2}(\pi)=\{1,4,8\}$, $\operatorname{DB}_{3}(\pi)=\{6,9\}$, $\operatorname{DB}_{4}(\pi)=\{2,7\}$, $\operatorname{DB}_{5}(\pi)=\{5\}$.

Lemma 3.3**.**

Fix integers $r,p,k,m$ and mutually disjoint subsets $C_{1},\dots,C_{p}$ of $\{1,2,\dots,m\}$ such that

[TABLE]

Let $u=|C_{1}|+\dots+|C_{p}|-k$. Then, for every $r$-tuples $B=(B_{1},\dots,B_{r})$ of mutually disjoint subsets of $\{1,2,\dots,k\}$ satisfying $B_{i}\subseteq C_{1}\cup\dots\cup C_{i}$ for $1\leq i\leq r$, we have

[TABLE]

Moreover, the equality holds if and only if the following conditions hold: $r=p$ and $B_{i}=C_{i}$ for $1\leq i\leq r-1$ and $B_{r}\subseteq C_{r}$.

Proof.

First assume that the conditions for the equality hold. Then

[TABLE]

Since $p=r$, we have $|B_{r}|=|C_{p}|-u$ and the equality of (3) holds.

Now suppose that $B=(B_{1},\dots,B_{r})$ is an $r$-tuple of mutually disjoint subsets of $\{1,2,\dots,k\}$ satisfying $B_{i}\subseteq C_{1}\cup\dots\cup C_{i}$ for $1\leq i\leq r$ that minimizes the value $\sum_{i=1}^{r}i|B_{i}|$. It suffices to show that $B$ satisfies the conditions for the equality. For a contradiction, suppose that the conditions do not hold. Then there are two cases.

Case 1: We can find the smallest $1\leq s\leq r-1$ such that $B_{s}\neq C_{s}$. By the assumption on $B$, there is an element $x\in B_{s}\cap C_{j}$ for some $1\leq j\leq s-1$. Define $B^{\prime}=(B^{\prime}_{1},\dots,B^{\prime}_{r})$ to be the $r$-tuple obtained from $B$ by replacing $B_{j}$ by $B_{j}\cup\{x\}$ and $B_{s}$ by $B_{s}\setminus\{x\}$. Then $\sum_{i=1}i|B^{\prime}_{i}|<\sum_{i=1}i|B_{i}|$, which is a contradiction.

Case 2: $\operatorname{DB}_{i}(\pi)\neq C_{i}$ for $1\leq i\leq r-1$ and $\operatorname{DB}_{r}(\pi)\not\subseteq C_{r}$. Similarly, we can find an element $x\in B_{r}\cap C_{j}$ for some $1\leq j\leq r-1$ and obtain a contradiction.

By the above two cases, $B$ must satisfy the conditions for the equality. This finishes the proof. ∎

The following proposition is the key ingredient for proving Chapoton’s conjecture.

Proposition 3.4**.**

Let $P$ be a naturally labeled poset on $\{1,2,\dots,m\}$. Suppose that $b_{1}\leq b_{2}\leq\dots\leq b_{m}$ is the increasing rearrangement of $\operatorname{mc}(1),\operatorname{mc}(2),\dots,\operatorname{mc}(m)$ and

[TABLE]

Then, for $\pi\in\mathcal{L}(P)$ and $0\leq k\leq m$, we have

[TABLE]

The equality holds if and only if all of the following conditions hold:

• •

$\operatorname{Des}(\pi)\subseteq\{1,2,\dots,k\}$,

• •

$\operatorname{DB}_{i}(\pi)=C_{i}$, for $1\leq i\leq p-1$,

• •

$\operatorname{DB}_{p}(\pi)\subseteq C_{p}$,

where $p$ is the integer satisfying

[TABLE]

Furthermore, for every $0\leq k\leq m$, there is a permutation in $\mathcal{L}(P)$ satisfying these conditions.

Proof.

For $\pi\in\mathfrak{S}_{m}$, let

[TABLE]

Suppose that $\pi$ is a permutation in $\mathcal{L}(P)$ such that $b(\pi)$ is the smallest. If $\pi$ has a descent greater than $k$, by Lemma 3.1, we can find $\sigma\in\mathcal{L}(P)$ with $\operatorname{Des}(\sigma)=\operatorname{Des}(\pi)\setminus\{\max(\operatorname{Des}(\pi))\}$. Then $b(\sigma)<b(\pi)$, which is a contradiction. If $k$ is the largest descent of $\pi$, by the same construction, we can remove the descent $k$ without changing $b(\pi)$. Therefore we can assume that all descents of $\pi$ are at most $k-1$.

Let $1\leq t_{1}<t_{2}<\dots<t_{r}=k$ be the non-descents of $\pi$ among $1,2,\dots,k$, i.e.,

[TABLE]

Then

[TABLE]

Note that $\operatorname{DB}_{i}(\pi)=\{\pi_{t_{i-1}+1},\dots,\pi_{t_{i}}\}$ for $1\leq i\leq r$, where $t_{0}=0$, and

[TABLE]

Since $t_{i}=|\operatorname{DB}_{1}(\pi)|+\dots+|\operatorname{DB}_{i}(\pi)|$ for $1\leq i\leq r$ and $k=|\operatorname{DB}_{1}(\pi)|+\dots+|\operatorname{DB}_{r}(\pi)|$, we have

[TABLE]

We claim that $\operatorname{mc}(x)\leq i$ for all $x\in\operatorname{DB}_{i}(\pi)$. To prove this let $\operatorname{mc}(x)=\ell$ and $x_{1}<_{P}\dots<_{P}x_{\ell}=x$ be a maximal chain ending at $x$. Since $\pi$ is a linear extension of $P$, $x_{1},\dots,x_{\ell}$ must occur in this order in $\pi$. Since $P$ is naturally labeled, we have $x_{1}<\dots<x_{\ell}$. Hence each $\operatorname{DB}_{j}(\pi)$ has at most one element among $x_{1},\dots,x_{\ell}$. This settles the claim.

The above claim implies that

[TABLE]

Let $u=|C_{1}|+\dots+|C_{p}|-k$. By Lemma 3.3, we have

[TABLE]

where the equality holds if and only if the following conditions hold: $r=p$ and $\operatorname{DB}_{i}(\pi)=C_{i}$ for $1\leq i\leq p-1$ and $\operatorname{DB}_{p}(\pi)\subseteq C_{p}$.

Now it remains to show that there is $\pi\in\mathcal{L}(P)$ satisfying the conditions for the equality. We construct such a permutation as follows. Let $T$ be any subset of $C_{p}$ such that $k=|C_{1}|+\dots+|C_{p-1}|+|T|$. Let $\pi=\pi_{1}\dots\pi_{m}\in\mathfrak{S}_{m}$ be the permutation obtained from the empty sequence by appending the elements of $C_{i}$ in decreasing order for $i=1,2,\dots,p-1$, the elements of $T$ in decreasing order, and the remaining integers in increasing order. Then $\pi$ satisfying the conditions for the equality. For a contradiction, suppose that $\pi\not\in\mathcal{L}(P)$. Then there are two elements $x,y\in P$ such that $x<_{P}y$ and $y$ appears to the left of $x$ in $\pi$. Let $\operatorname{mc}(x)=i$ and $\operatorname{mc}(y)=j$. Then $x\in C_{i}$ and $y\in C_{j}$. Since $x<_{P}y$, we have $i<j$ and $x<y$. If $y$ is in $\pi_{1}\dots\pi_{k}$, then $j\leq p$. In this case we have $x\in C_{i}=\operatorname{DB}_{i}(\pi)$ and $y\in C_{j}=\operatorname{DB}_{j}(\pi)$, which is a contradiction to the assumption that $y$ appear to the left of $x$. If $y$ is not in $\pi_{1}\dots\pi_{k}$, we have both $x$ and $y$ in $\pi_{k+1}\dots\pi_{m}$. Since these elements are in increasing order, we cannot have $y$ to the left of $x$. Therefore we must have $\pi\in\mathcal{L}(P)$, which finishes the proof. ∎

The following corollary is an immediate consequence of Proposition 3.4.

Corollary 3.5**.**

Let $P$ be a naturally labeled poset on $\{1,2,\dots,m\}$. Suppose that $b_{1}\leq b_{2}\leq\dots\leq b_{m}$ is the increasing rearrangement of $\operatorname{mc}(1),\operatorname{mc}(2),\dots,\operatorname{mc}(m)$. Then, for $0\leq k\leq m$, we have

[TABLE]

Moreover, for $1\leq k\leq m$, if $P$ is not a chain, we have

[TABLE]

Note that Corollary 3.5 allows us to find the minimum of $\operatorname{maj}(\pi)-k\operatorname{des}(\pi)$ over $\pi\in\mathcal{L}(P)$. The second part of Corollary 3.5 means that if $1\leq k\leq m$ and $P$ is not a chain, the minimum of $\operatorname{maj}(\pi)-k\operatorname{des}(\pi)+\binom{k+1}{2}$ for all $\pi\in\mathcal{L}(P)$ is attained for $\pi\in\mathcal{L}(P)$ satisfying $1\leq\operatorname{des}(\pi)\leq k$. This will be used in the next section.

4. Proof of Theorem 2.5

In this section we assume that $P$ is a naturally labeled poset on $\{1,2,\dots,m\}$ and $b_{1}\leq b_{2}\leq\dots\leq b_{m}$ is the increasing rearrangement of $\operatorname{mc}(1),\dots,\operatorname{mc}(m)$.

For a polynomial $f(q)$ in $q$, define

[TABLE]

When $f(q)=0$, we use the following convention:

[TABLE]

Recall that

[TABLE]

Since $P$ is naturally labeled, $\mathcal{L}(P)$ contains the identity permutation. Therefore,

[TABLE]

where

[TABLE]

Since $[x^{0}]F_{P}(q,x)=[x^{0}]A=[m]_{q}!$, we have

[TABLE]

Therefore, in order to prove Theorem 2.5, it suffices to show the following two propositions.

Proposition 4.1**.**

For $1\leq k\leq m$, we have

[TABLE]

Proposition 4.2**.**

For $1\leq k\leq m$, we have

[TABLE]

Proof of Proposition 4.1.

By (8), it is enough to show that

[TABLE]

[TABLE]

In order to get the largest power of $q$, when we expand the product in $A$, we must select $q^{i}x$ or $q^{i-1}$. This implies (9).

To prove (10), consider $\pi\in\mathfrak{S}_{m}$ with $\operatorname{des}(\pi)=s\geq 1$. Then

[TABLE]

Therefore, we obtain (10). ∎

The rest of this section is devoted to proving Proposition 4.2.

For $\pi\in\mathfrak{S}_{m}$ with $\operatorname{des}(\pi)=s\geq 1$ and an integer $1\leq k\leq m$, let

[TABLE]

Then we always have

[TABLE]

We need the following two lemmas.

Lemma 4.3**.**

Let $\pi\in\mathfrak{S}_{m}$ with $\operatorname{des}(\pi)=s\geq 1$. Then, for $s\leq k\leq m$, we have

[TABLE]

Proof.

It is easy to see that the smallest power of $q$ in $[x^{k-1}]t(\pi,k)$ is obtained if and only if we select $x^{s-1}$ in the first product and $q^{1+2+\dots+(k-s)}x^{k-s}$ in the second product in (11). Thus, we obtain

[TABLE]

which is equivalent to the lemma. ∎

Lemma 4.4**.**

Let $P$ be a naturally labeled poset on $\{1,2,\dots,m\}$. Suppose that $P$ is not a chain. Then, for $1\leq k\leq m$, we have

[TABLE]

Proof.

Let

[TABLE]

By (12), for $\pi\in\mathcal{L}(P)$ with $\operatorname{des}(\pi)\geq k+1$, we have

[TABLE]

By Lemma 3.1, we can find $\sigma\in\mathcal{L}(P)$ such that $\operatorname{Des}(\sigma)=\operatorname{Des}(\pi)\setminus\{\max(\operatorname{Des}(\pi))\}$. Since $\max(\operatorname{Des}(\pi))\geq\operatorname{des}(\pi)$, we have

[TABLE]

By repeating this argument, we obtain that

[TABLE]

for some $\tau\in\mathcal{L}(P)$ with $\operatorname{des}(\tau)=k$.

On the other hand, by Lemma 4.3, we have

[TABLE]

By (13) and (14), we have $m_{1}<m_{2}$. Therefore $q_{\min}([x^{k}]B)=m_{1}$. By applying Lemma 4.3 to $m_{1}$, we obtain the desired identity. ∎

Now we give a proof of Proposition 4.2.

Proof of Proposition 4.2.

First, observe that

[TABLE]

If $P$ is a chain, then the identity permutation is the only linear extension of $P$. In this case $B=0$ and $b_{i}=i$. Thus

[TABLE]

Now suppose that $P$ is not a chain. By Lemma 4.4 and Corollary 3.5 we have

[TABLE]

Therefore we also obtain

[TABLE]

Acknowledgement

The authors would like to thank the anonymous referee for his or her careful reading and helpful comments.