Whitney Numbers for Poset Cones

TL;DR

This paper explores Whitney numbers within cones of the braid arrangement, linking them to poset linear extensions and refining counts via permutation statistics, providing new combinatorial interpretations and generating functions.

Contribution

It introduces a new interpretation of Whitney numbers for cones in the braid arrangement as counts of linear extensions refined by permutation statistics.

Findings

Whitney numbers refine the count of linear extensions of posets.

A permutation statistic generalizes the number of left-to-right maxima.

A generating function for Whitney numbers in disjoint union of chains is derived.

Abstract

Hyperplane arrangements dissect $\mathbb{R}^n$ into connected components called chambers, and a well-known theorem of Zaslavsky counts chambers as a sum of nonnegative integers called Whitney numbers of the first kind. His theorem generalizes to count chambers within any cone defined as the intersection of a collection of halfspaces from the arrangement, leading to a notion of Whitney numbers for each cone. This paper focuses on cones within the braid arrangement, consisting of the reflecting hyperplanes $x_i=x_j$ inside $\mathbb{R}^n$ for the symmetric group, thought of as the type $A_{n-1}$ reflection group. Here cones correspond to posets, chambers within the cone correspond to linear extensions of the poset, and the Whitney numbers of the cone interestingly refine the number of linear extensions of the poset. We interpret this refinement for all posets as counting linear extensions according to a statistic that generalizes the number of left-to-right maxima of a permutation. When the poset is a disjoint union of chains, we interpret this refinement differently, using Foata's theory of cycle decomposition for multiset permutations, leading to a simple generating function compiling these Whitney numbers.

Equations

$$[V,X]:=\{Y\in \mathcal{L}(\mathcal{A}):V\le Y\le X\}$$

$$\#\mathcal{C}(\mathcal{A})=\sum _{X\in \mathcal{L}(\mathcal{A})}|\mu (V,X)|=\sum _{k=0}^n{c}_k(\mathcal{A})={\left[\mathsf{Poin}(\mathcal{A},t)\right]}_{t=1},$$

$$c_k(\mathcal{A}):=\sum _{\begin{array}{c}X\in \mathcal{L}(\mathcal{A}):\\ \mathrm{codim}(X)=k\end{array}}|\mu (V,X)|,$$

$$\mathsf{Poin}(\mathcal{A},t):=\sum _{k\ge 0}{c}_k(\mathcal{A})t^k$$

$${\mathcal{L}}^{\mathrm{int}}(\mathcal{K})=\{X\in \mathcal{L}(\mathcal{A})\mid X\cap \mathcal{K}\ne \varnothing \}.$$

$$\#\mathcal{C}(\mathcal{K})=\sum _{X\in {\mathcal{L}}^{\mathrm{int}}(\mathcal{K})}|\mu (V,X)|=\sum _{k=0}^n{c}_k(\mathcal{K})={\left[\mathsf{Poin}(\mathcal{K},t)\right]}_{t=1}.$$

$$c_k(\mathcal{K}):=\sum _{\begin{array}{c}X\in {\mathcal{L}}^{\mathrm{int}}(\mathcal{K}):\\ \mathrm{codim}(X)=k\end{array}}|\mu (V,X)|,$$

$$\begin{array}[]{ccccccc}&&{\sf Poin}(\mathcal{K},t)&=&c_{0}(\mathcal{K})+c_{1}(\mathcal{K})t+c_{2}(\mathcal{K})t^{2}&=&1+3t+t^{2},\\ \#\mathcal{C}({\mathcal{K}})&=&\left[{\sf Poin}(\mathcal{K},t)\right]_{t=1}&=&c_{0}(\mathcal{K})+c_{1}(\mathcal{K})+c_{2}(\mathcal{K})&=&5.\end{array}$$

$$H_{ij}=\{(x_1,\dots ,x_n)\in V={\mathbb{R}}^n\mid x_i-x_j=0\}$$

$${\mathcal{K}}_{\sigma }:=\{x\in V={\mathbb{R}}^n:x_{{\sigma }_1}<x_{{\sigma }_2}<\cdots <x_{{\sigma }_n}\}.$$

$${\mathcal{K}}_P:=\{x\in V={\mathbb{R}}^n:x_i<x_j\text{ for }i{<}_{P}j\}.$$

$$\#\mathrm{LinExt}(P)=\sum _{k\ge 0}{c}_k(P)={\left[\mathsf{Poin}(P,t)\right]}_{t=1}$$

$$n!=\#{\mathfrak{S}}_n=\sum _{k\ge 0}c(n,k).$$

$$\mathsf{Poin}({\mathsf{Antichain}}_n,t)=\sum _{k\ge 0}c(n,k)t^{n-k}$$

$$=\sum _{\sigma \in {\mathfrak{S}}_n}{t}^{n-\mathsf{LRmax}(\sigma )}$$

$$=1(1+t)(1+2t)\cdots (1+(n-1)t).$$

$$\mathsf{Poin}(P,t)=\sum _{\sigma \in {\mathfrak{S}}^{⋔}(P)}{t}^{n-\mathsf{cyc}(\sigma )}.$$

$$\begin{array}[]{rcl}\mathfrak{S}^{\pitchfork}(P)&\overset{\Phi}{\longrightarrow}&\operatorname{LinExt}(P)\\ \tau&\longmapsto&\sigma\end{array}$$

$$\mathsf{Poin}(P,t)=\sum _{\sigma \in \mathrm{LinExt}(P)}{t}^{n-{\mathsf{LRmax}}_P(\sigma )}.$$

$$P_{\overline{a}}:={\mathsf{a}}_1\bigsqcup {\mathsf{a}}_2\bigsqcup \cdots \bigsqcup {\mathsf{a}}_{\ell }$$

$$\begin{array}[]{rcl}\operatorname{LinExt}(P_{\overline{a}})&\longrightarrow&\mathfrak{S}^{\pitchfork}(P_{\overline{a}})\\ \sigma&\longmapsto&\tau\end{array}$$

$$\mathsf{Poin}(P_{\overline{a}},t)=\sum _{\sigma \in \mathrm{LinExt}(P_{\overline{a}})}{t}^{n-\mathsf{pcyc}(\sigma )}.$$

$$\sum _{\overline{a}\in \{1,2,\dots {\}}^{\ell }}\mathsf{Poin}(P_{\overline{a}},t)\cdot {\mathbf{x}}^{\overline{a}}=\frac{1}{1-{\sum }_{j=1}^{\ell }{e}_j(\mathbf{x})\cdot (t-1)(2t-1)\cdots ((j-1)t-1)},$$

$${\mathsf{des}}_{P_1,P_2}(\sigma ):=\#\{i\in [n-1]:{\sigma }_i\in P_2,{\sigma }_{i+1}\in P_1,\text{ with }{\sigma }_i,{\sigma }_{i+1}\text{ incomparable in }P\}.$$

$$\mathsf{Poin}(P,t)=\sum _{\sigma \in \mathrm{LinExt}(P)}{t}^{{\mathsf{des}}_{P_1,P_2}(\sigma )}.$$

$$\sum _{\sigma \in \mathrm{LinExt}(P)}{t}^{\mathsf{des}(\sigma )}$$

$$\mathsf{des}(\sigma ):=\#\{i\in [n-1]:{\sigma }_i>{\sigma }_{i+1}\},$$

$$H_{ij}=\{(x_1,\dots ,x_n)\in {\mathbb{R}}^n\mid x_i-x_j=0\}$$

$${\mathcal{K}}_{1243}$$

$${\mathcal{K}}_{4213}$$

$$\bigcap _{H\in \varnothing }H=V={\mathbb{R}}^n.$$

$$\mu (V,X_{\pi })=(-1{)}^{n-k}\prod _{i=1}^k(\#B_i-1)!\left(=\mu (1|2|\cdots |n,\pi )\right)$$

Full text

• •

cones correspond to posets,

• •

chambers within the cone correspond to linear extensions of the poset,

• •

the Whitney numbers of the cone interestingly refine the number of linear extensions of the poset.

We interpret this refinement for all posets as counting linear extensions according to a statistic that generalizes the number of left-to-right maxima of a permutation. When the poset is a disjoint union of chains, we interpret this refinement differently, using Foata’s theory of cycle decomposition for multiset permutations, leading to a simple generating function compiling these Whitney numbers.

The first and third authors were partially supported by NSF grant DMS-1601961. The second author was supported by NRF grants #2019R1F1A1059081 and #2016R1A5A1008055.

1. Introduction

This paper concerns arrangements $\mathcal{A}=\{H_{1},\ldots,H_{m}\}$ of hyperplanes $H_{i}$, which are affine-linear codimension one subspaces of $V=\mathbb{R}^{n}$. Each such arrangement dissects $V$ into the connected components of its complement $V\setminus\bigcup_{i=1}^{m}H_{i}$, called chambers. Let $\mathcal{C}({\mathcal{A}})$ denote the collection of all such chambers.

The theory of hyperplane arrangements is rich and well-explored, with connections to reflection groups, braid groups, random walks, card-shuffling, and discrete geometry of polytopes and oriented matroids; see [19, 27]. In particular, the number of chambers $\#\mathcal{C}({\mathcal{A}})$ has a famous formula due to Zaslavsky [31], expressed in terms of the intersection poset $\mathcal{L}(\mathcal{A})$, consisting of all intersection subspaces $X=H_{i_{1}}\cap H_{i_{2}}\cap\cdots\cap H_{i_{k}}$, ordered via reverse inclusion. This poset is known to have the property that every lower interval

[TABLE]

from its unique bottom element $V$ to any intersection space $X$ forms a geometric lattice. In particular, each such $[V,X]$ is a ranked poset, where the rank of $X$ is $\operatorname{codim}(X):=n-\dim(X)$. Zaslavsky’s result says that

[TABLE]

where $\mu(-,-)$ denotes the Möbius function of $\mathcal{L}(\mathcal{A})$, while the nonnegative integers

[TABLE]

are often called the (signless) Whitney numbers of the first kind for $\mathcal{A}$, and their generating function

[TABLE]

is called the Poincaré polynomial111This is because it is the generating function for the Betti numbers of the complexified complement $\mathbb{C}^{n}\setminus\mathcal{A}$; see [19, Chap. 5].

Our starting point is a less widely-known generalization of equation (1), also due to Zaslavsky [32]. It applies more generally to count the chambers of $\mathcal{A}$ that lie within a cone $\mathcal{K}$, defined to be the intersection of any collection of open halfspaces for hyperplanes of $\mathcal{A}$; said differently, a cone $\mathcal{K}$ of $\mathcal{A}$ is a chamber in $\mathcal{C}({\mathcal{A}^{\prime}})$ for some subarrangement $\mathcal{A}^{\prime}\subseteq\mathcal{A}$. Results on the set $\mathcal{C}({\mathcal{K}})$ of all chambers of $\mathcal{A}$ inside a cone $\mathcal{K}$ have appeared more recently in work of Brown on random walks [8], and in work of Gente on Varchenko determinants [13, Section 2.4], and independently Aguiar and Mahajan [1, Theorem 8.22]. Define the poset of interior intersections for $\mathcal{K}$ to be the following order ideal in $\mathcal{L}(\mathcal{A})$:

[TABLE]

Zaslavsky observed in [32, Example A, p. 275] that (1) generalizes to cones $\mathcal{K}$, asserting

[TABLE]

Here we again define nonnegative integers, the (signless) Whitney numbers of the first kind for the cone $\mathcal{K}$

[TABLE]

with generating function ${\sf Poin}(\mathcal{K},t):=\sum_{k}\thinspace c_{k}(\mathcal{K})\thinspace t^{k}$, which we call the Poincaré polynomial of $\mathcal{K}$.

For example, inside $\mathcal{A}=\{H_{1},H_{2},H_{3},H_{4},H_{5}\}$ in $\mathbb{R}^{2}$ shown below on the left, we have shaded one of four possible cones $\mathcal{K}$ defined by the subarrangement $\mathcal{A}^{\prime}=\{H_{4},H_{5}\}$, containing $\#\mathcal{C}({\mathcal{K}})=5$ chambers of $\mathcal{A}$:

H_{4}$$H_{1}$$H_{2}$$H_{3}$$H_{5}

H_{4}$$H_{1}$$H_{2}$$H_{3}$$H_{5}

Zaslavsky’s formula (2) computes this as follows. The poset of interior intersections $\mathcal{L}^{\operatorname{int}}(\mathcal{K})$ has Hasse diagram:

V$$H_{1}$$H_{2}$$H_{3}$$H_{2}\cap H_{3}

Here $\mu(V,X)=(-1)^{\operatorname{codim}(X)}$ for all $X$, so that $(c_{0}(\mathcal{K}),c_{1}(\mathcal{K}),c_{2}(\mathcal{K}))=(1,3,1)$, and

[TABLE]

The object of this paper is to understand the distribution of the signless Whitney numbers as a refinement of $\#\mathcal{C}({\mathcal{K}})$ as in equation (2), for cones $\mathcal{K}$ in the braid arrangement. The braid arrangement $A_{n-1}=\{H_{ij}\}_{1\leq i<j\leq n}$ is the set of $\binom{n}{2}$ reflecting hyperplanes

[TABLE]

for the symmetric group $\mathfrak{S}_{n}$ on $n$ letters, thought of as the reflection group of type $A_{n-1}$. There is a well-known and easy bijection between the chambers $\mathcal{C}({A_{n-1}})$ and the permutations $\sigma=[\sigma_{1},\sigma_{2},\ldots,\sigma_{n}]$ in $\mathfrak{S}_{n}$, sending $\sigma$ to the chamber:

[TABLE]

More generally, one has an easy bijection, reviewed in Section 2, between posets on the set $[n]:=\{1,2,\ldots,n\}$ and cones in the braid arrangement $A_{n-1}$, sending a poset $P$ to the cone

[TABLE]

It is readily checked that the chamber $\mathcal{K}_{\sigma}$ lies in the cone $\mathcal{C}({\mathcal{K}_{P}})$ if and only if $\sigma$ is a linear extension of $P$, meaning that $i<_{P}j$ implies $i<_{\sigma}j$, regarding $\sigma$ as a total order $\sigma_{1}<\sigma_{2}<\cdots<\sigma_{n}$ on $[n]$. Letting $\operatorname{LinExt}(P)$ denote the set of all linear extensions of $P$, this shows that $\#\mathcal{C}({\mathcal{K}_{P}})=\#\operatorname{LinExt}(P)$, and hence equation (2) becomes

[TABLE]

abbreviating $\thinspace c_{k}(P)\thinspace:=\thinspace c_{k}(\mathcal{K}_{P})\thinspace$ and ${\sf Poin}(P,t):={\sf Poin}(\mathcal{K}_{P},t)$ from now on.

A motivating special case occurs when $P={\sf Antichain}_{n}$ is the antichain poset on $[n]$ that has no order relations. In this case $\operatorname{LinExt}({\sf Antichain}_{n})=\mathfrak{S}_{n}$ itself, and the signless Whitney number $\thinspace c_{n-k}({\sf Antichain}_{n})\thinspace$ of the first kind is well-known [28, Prop. 1.3.7] to be the signless Stirling number of the first kind $c(n,k)$ that counts the permutations in $\mathfrak{S}_{n}$ having $k$ cycles. Consequently, (4) becomes the easy summation formula

[TABLE]

Our main results will generalize three well-known expressions for ${\sf Poin}({\sf Antichain}_{n},t)$, explained below:

[TABLE]

Equation (6) comes from the definition of $c(n,k)$ mentioned above, where $\mathsf{cyc}(\sigma)$ is the number of cycles of $\sigma$. This interpretation of the Poincaré polynomial will be generalized to all posets in Theorem 1.1, and generalized in a different way to posets which are disjoint unions of chains in Theorem 1.3.

Equation (7) arises from a well-known bijection $\tau\mapsto\sigma$ from $\mathfrak{S}_{n}$ to itself, called the fundamental bijection222This bijection, often called Foata’s first fundamental transformation, seems to have appeared also in work of Rényi [24, §4]., such that $\mathsf{cyc}(\tau)={\sf LRmax}(\sigma)$, see [28, Proposition 1.3.1]; here ${\sf LRmax}(\sigma)$ is the number of left-to-right maxima of $\sigma=\sigma_{1}\sigma_{2}\dots\sigma_{n}$, i.e., the number of integers $1\leq j\leq n$ such that $\sigma_{j}=\max\{\sigma_{1},\sigma_{2},\dots,\sigma_{j}\}$. This bijection and interpretation of the Poincaré polynomial will be generalized to all posets in Theorem 1.2.

Equation (8) is well-known [28, Proposition 1.3.7], and will be generalized to a generating function compiling the Poincaré polynomials of all posets which are disjoint unions of chains in Theorem 1.4 below.

The remainder of this paper is organized as follows.

Section 2 gives preliminaries on the intersection lattice and cones in braid arrangements. In particular, for a poset $P$ on $[n]$, it gives an explicit combinatorial description of the interior intersections $\mathcal{L}^{\operatorname{int}}(\mathcal{K}_{P})$ for a poset cone $\mathcal{K}_{P}$, as the subset $\Pi^{\pitchfork}(P)$ of $P$-transverse partitions inside the lattice $\Pi_{n}$ of set partitions of $[n]$; roughly speaking, these are the set partitions $\pi$ for which the quotient preposet $P/\pi$ does not collapse any strict order relations $i<_{P}j$ into equalities. With this in hand, it defines the subset $\mathfrak{S}^{\pitchfork}(P)$ of $P$-transverse permutations inside $\mathfrak{S}_{n}$ as those permutations $\sigma$ whose cycles form a $P$-transverse partition, in order to prove the following generalization of equation (6).

Theorem 1.1**.**

For any poset $P$ on $[n]$, one has

[TABLE]

In particular, $\#\operatorname{LinExt}(P)=\#\mathfrak{S}^{\pitchfork}(P)$.

This result even generalizes to a statement (Theorem 2.30) about cones in any real reflection arrangement.

In light of Theorem 1.1, one expects bijections between $\mathfrak{S}^{\pitchfork}(P)$ and $\operatorname{LinExt}(P)$. One such bijection is the goal of Section 3, which defines a notion of $P$-left-to-right maximum for a linear extension of $P$, and then generalizes the fundamental bijection $\tau\mapsto\sigma$ and equation (7) as follows.

Theorem 1.2**.**

For any poset $P$ on $[n]$, one has a bijection

[TABLE]

such that $\mathsf{cyc}(\tau)={\sf LRmax}_{P}(\sigma)$, where ${\sf LRmax}_{P}(\sigma)$ is the number of $P$-left-to-right maxima of $\sigma$. Therefore,

[TABLE]

Section 4 examines posets $P$ which are disjoint unions of chains, and produces a second interesting bijection $\mathfrak{S}^{\pitchfork}(P)\to\operatorname{LinExt}(P)$. Given any composition $\overline{a}=(a_{1},\ldots,a_{\ell})$ of $n$, meaning that $\overline{a}\in\{0,1,2,\ldots\}^{\ell}$ and $|\overline{a}|:=\sum_{i=1}^{\ell}a_{i}=n$, let $\mathsf{a_{i}}$ denote a chain (totally ordered) poset on $a_{i}$ elements, and then

[TABLE]

is a disjoint union of incomparable chains of sizes $a_{1},a_{2},\ldots,a_{\ell}$. Here one can identify elements in $\operatorname{LinExt}(P_{\overline{a}})$ with multiset permutations of $\{1^{a_{1}},2^{a_{2}},\dots,\ell^{a_{\ell}}\}$. Section 4 reviews the beautiful theory of prime cycle decompositions for such multiset permutations due to Foata [9], and then reinterprets it as giving a bijection $\operatorname{LinExt}(P_{\overline{a}})\rightarrow\mathfrak{S}^{\pitchfork}(P_{\overline{a}})$, and the following (second) generalization of equation (6).

Theorem 1.3**.**

For any composition $\overline{a}$ of $n$, the disjoint union $P_{\overline{a}}$ of chains has a bijection

[TABLE]

with $\mathsf{cyc}(\tau)=\mathsf{pcyc}(\sigma)$, the number of prime cycles in Foata’s unique decomposition for $\sigma$. Thus

[TABLE]

Foata’s theory is then used to prove the following generating function, generalizing equation (8).

Theorem 1.4**.**

For $\ell=1,2,\ldots$, one has

[TABLE]

where $\mathbf{x}^{\overline{a}}:=x_{1}^{a_{1}}\cdots x_{\ell}^{a_{\ell}}$ and $e_{j}(\mathbf{x}):=\sum_{1\leq i_{1}<\cdots<i_{j}\leq\ell}x_{i_{1}}\cdots x_{i_{j}}$ is the $j^{th}$ elementary symmetric function.

Section 5 then examines posets of width two, that is, posets $P$ decomposable as $P=P_{1}\cup P_{2}$ where the subposets $P_{1},P_{2}$ are chains (i.e. totally ordered subsets) inside $P$. Here the Whitney numbers $\thinspace c_{k}(P)\thinspace$ are interpreted by a descent-like statistic on $\sigma$ in $\operatorname{LinExt}(P)$:

[TABLE]

Theorem 1.5**.**

For a width two poset decomposed into two chains as $P=P_{1}\cup P_{2}$, one has

[TABLE]

Theorem 1.5 will be used to show (Corollary 5.3) that, for certain very special width two posets $P$, the Poincaré polynomial ${\sf Poin}(P,t)$ coincides with the $P$-Eulerian polynomial,

[TABLE]

which counts linear extensions $\sigma$ of $P$ according to their number of (usual) descents

[TABLE]

assuming that $P$ has been naturally labeled in the sense that the identity permutation $\sigma=[1,2,\dots,n]$ lies in $\operatorname{LinExt}(P)$. In particular, Example 5.4 uses this to deduce that for $P=\mathsf{2}\times\mathsf{n}$, the Cartesian product of chains having sizes $2$ and $n$, the Whitney numbers $\thinspace c_{k}(\mathsf{2}\times\mathsf{n})\thinspace$ are Narayana numbers, counting $2\times n$ standard tableaux according to their number of descents. This $P$-Eulerian polynomial has many interpretations, e.g., as the $h$-polynomial of the order complex for the distributive lattice $J(P)$ of order ideals in $P$, or of the $P$-partition triangulation of the order polytope for $P$; see [23, Proposition 2.1, Proposition 2.2] and [28, Sections 3.4, 3.8, 3.13] for more on this. Unfortunately, in general the $P$-Eulerian polynomial differs from the Poincaré polynomial ${\sf Poin}(P,t)$ considered here. For example, when $P$ is an antichain with three elements, the $P$-Eulerian polynomial is $1+4t+t^{2}$, while ${\sf Poin}(P,t)=1+3t+2t^{2}$.

Lastly, we note that the cone $\mathcal{K}_{P}$ associated to a poset $P$ has appeared in many places in the literature, for example, implicitly in Stanley’s theory of $P$-partitions [28, §3.15] and related work of Gessel on quasisymmetric functions [12], more explicitly in work of Björner and Wachs [4, 5, 6], and as the ranking COMs of Bandelt, Chepoi and Knauer [2]. A pointed version of this cone, called an order cone is discussed in Beck and Sanyal [3, Chap. 6]. When the underlying Hasse diagram of $P$ is a tree, results on the number of chambers $\#\mathcal{C}({\mathcal{K}_{P}})=\#\operatorname{LinExt}(P)$ in these cones appear in work of K. Saito [25].

2. The braid arrangement, its intersection lattice, and its cones

2.1. Preliminaries on arrangements

We begin with some preliminaries on hyperplane arrangements, focusing on braid arrangements. Good references include [19], [27], [28, §3.11], [20, §3.3].

Definition 2.1**.**

A hyperplane in $V=\mathbb{R}^{n}$ is an affine linear subspace of codimension one. An arrangement of hyperplanes in $\mathbb{R}^{n}$ is a finite collection $\mathcal{A}=\{H_{1},\dots,H_{m}\}$ of distinct hyperplanes. A chamber of $\mathcal{A}$ is an open, connected component of $\mathbb{R}^{n}\backslash\bigcup_{H\in\mathcal{A}}H$. The set of all chambers of $\mathcal{A}$ is denoted by $\mathcal{C}({\mathcal{A}})$.

Example 2.2**.**

The type A reflection arrangement, $A_{n-1}$, also called the braid arrangement, consists of the $\binom{n}{2}$ hyperplanes of the form

[TABLE]

for integers $1\leq i<j\leq n$. There are $n!$ chambers $\mathcal{K}_{\sigma}=\{x\in\mathbb{R}^{n}:x_{\sigma_{1}}<\cdots<x_{\sigma_{n}}\}$ of $A_{n-1}$, naturally indexed by the permutations $\sigma=[\sigma_{1},\sigma_{2},\ldots,\sigma_{n}]$ of $[n]$, that give the strict inequalities ordering the coordinates within the chamber, as in (3). For example, when $n=4$,

[TABLE]

are two out of the $4!=24$ chambers of $\mathcal{C}({A_{4-1}})$.

Definition 2.3**.**

Let $\mathcal{A}$ be a hyperplane arrangement in $\mathbb{R}^{n}$. An intersection of $\mathcal{A}$ is a nonempty subspace of the form $X=H_{i_{1}}\cap H_{i_{2}}\cap\cdots\cap H_{i_{k}}$ where $\{H_{i_{1}},H_{i_{2}},\dots,H_{i_{k}}\}\subseteq\mathcal{A}$. Here the ambient vector space $V=\mathbb{R}^{n}$ is considered to be the intersection $\bigcap_{H\in\varnothing}H$ of the empty set of hyperplanes. We denote the set of intersections of $\mathcal{A}$ by $\mathcal{L}(\mathcal{A})$.

Example 2.4**.**

The intersections of $A_{n-1}$ are described by equalities between the variables.

• •

For all $n\geq 1$, the line $x_{1}=x_{2}=\dots=x_{n}$ is the intersection of all the hyperplanes of $A_{n-1}$.

• •

When $n=4$ the intersection of $H_{12}$ and $H_{34}$ in the subspace of $\mathbb{R}^{4}$ in which $x_{1}=x_{2}$ and $x_{3}=x_{4}$. On the other hand, the intersection of $H_{12}$ and $H_{13}$ is the subspace of $\mathbb{R}^{4}$ in which $x_{1}=x_{2}=x_{3}$.

More generally, there is a bijection $\pi\mapsto X_{\pi}$ between the collection $\Pi_{n}$ of all set partitions $\pi=\{B_{1},\dots,B_{k}\}$ of $[n]=\{1,2,\dots,n\}$ and the set of all intersections of $A_{n-1}$. The bijection sends the set partition $\pi$ to the subspace $X_{\pi}$ where one has equal coordinates $x_{i}=x_{j}$ whenever $i,j$ lie in a common block $B_{k}$ of $\pi$. We sometimes denote the set partition $\pi=\{B_{1},\dots,B_{k}\}$ with the notation $\pi=B_{1}|B_{2}|\cdots|B_{k}$, and may or may not include commas and set braces around the elements of each block $B_{i}$. E.g., $1\mid 23\mid 456$ and $\{\{1\},\{2,3\},\{4,5,6\}\}$ represent the same set partition of $[6]$.

• •

For example, the set partition $1|2|\cdots|n$ in which all elements appear as singletons corresponds to $X_{1|2|\cdots|n}=V=\mathbb{R}^{n}$, the empty intersection, which is the ambient space.

• •

For all $n\geq 1$, the set partition $123\cdots n$ having all the elements in the same block corresponds to the line $X_{123\cdots n}$ defined by $x_{1}=x_{2}=\dots=x_{n}$.

• •

When $n=4$, one has $X_{12|34}=H_{12}\cap H_{34}$ and $X_{123|4}=H_{12}\cap H_{13}$.

The collection $\mathcal{L}(\mathcal{A})$ of all intersections of an arrangement $\mathcal{A}$ will be partially ordered by reverse inclusion, and called the intersection poset of $\mathcal{A}$. It has a unique minimal element, namely the intersection

[TABLE]

Let $\Pi_{n}$ denote the lattice of set partitions on $[n]$, ordered via refinement: $\pi_{1}\leq\pi_{2}$ if $\pi_{1}$ refines $\pi_{2}.$ For the braid arrangement $A_{n-1}$, the intersection poset $\mathcal{L}(A_{n-1})$ is isomorphic to $\Pi_{n}$.

Proposition 2.5** ([27, pp. 26-27]).**

The map $\pi\mapsto X_{\pi}$ in Example 2.4 is a poset isomorphism $\Pi_{n}\cong\mathcal{L}(A_{n-1}).$

For any hyperplane arrangement $\mathcal{A}$, each of the lower intervals $[V,X]:=\{Y\in\mathcal{L}(\mathcal{A}):V\leq Y\leq X\}$ forms a geometric lattice [28, Prop. 3.11.2]. In particular, this implies that each such lower interval is a ranked poset, with rank function given by the codimension $\operatorname{codim}(X)=\dim(V)-\dim(X)$. Furthermore, this implies that its Möbius function values $\mu(V,X)$, defined recursively by $\mu(V,V):=1$ and $\mu(V,X):=-\sum_{Y:V\leq Y<X}\mu(V,Y)$, will alternate in sign in the sense that $(-1)^{\operatorname{codim}(X)}\mu(V,X)\geq 0$.

For the braid arrangement, these Möbius function values have a simple expression.

Proposition 2.6** ([28, Example 3.10.4]).**

For any set partition $\pi=\{B_{1},\dots,B_{k}\}$ in $\Pi_{n}$, one has

[TABLE]

with the convention $0!:=1$. Here $\mu(V,X_{\pi}),\mu(1|2|\cdots|n,\pi)$ are $\mu(-,-)$ values in $\mathcal{L}(A_{n-1}),\Pi_{n}$, respectively.

For the sake of our later discussion, we point out a re-interpretation of this formula involving permutations. Given a permutation $\sigma$ in $\mathfrak{S}_{n}$, when considered as acting on $V=\mathbb{R}^{n}$, its fixed space $V^{\sigma}:=\{\mathbf{x}\in\mathbb{R}^{n}:\sigma(\mathbf{x})=\mathbf{x}\}$ will be the intersection subspace $V^{\sigma}=X_{\pi}$, where $\pi=\{B_{1},B_{2},\ldots,B_{k}\}$ is the set partition in $\Pi_{n}$ given by the cycles of $\sigma$. Consequently, one can re-interpret

[TABLE]

since each block $B_{i}$ of $\pi$ has $(\#B_{i}-1)!$ choices of a cyclic orientation.

Definition 2.7**.**

Let $\mathcal{A}$ be an arrangement of hyperplanes in $\mathbb{R}^{n}$. For $0\leq k\leq n$, the $k$th signless Whitney number of $\mathcal{L}(\mathcal{A})$ of the first kind is

[TABLE]

Henceforth, we call $c_{k}(\mathcal{A})$, $0\leq k\leq n$, the Whitney numbers of $\mathcal{A}$. One of the standard ways to compile them into a generating function is their Poincaré polynomial ${\sf Poin}(\mathcal{A},t):=\sum_{k=0}^{n}\thinspace c_{k}(\mathcal{A})\thinspace t^{k}$; see [19, §2.3].

As mentioned in the Introduction, we aim to understand the chambers, intersections, and Whitney numbers for cones in $\mathcal{A}$; the chambers, intersections, and Whitney numbers for $\mathcal{A}$ are a special case.

Definition 2.8**.**

Let $\mathcal{A}$ be an arrangement of hyperplanes in $V=\mathbb{R}^{n}$. A cone333Aguiar and Mahajan [1] call these objects top-cones. $\mathcal{K}$ of $\mathcal{A}$ is any nonempty intersection $\varnothing\neq\mathcal{K}\subseteq V=\mathbb{R}^{n}$ of (open) halfspaces defined by a subset $\mathcal{A}^{\prime}$ of the hyperplanes from $\mathcal{A}$. That is, a cone $\mathcal{K}$ is any one of the (open) chambers from the set of all chambers $\mathcal{C}({\mathcal{A}^{\prime}})$ for some subarrangement $\mathcal{A}^{\prime}\subseteq\mathcal{A}$. For example, in the following arrangement in $\mathbb{R}^{2}$ there are four cones defined by the dashed hyperplanes. One such cone $\mathcal{K}$ is shaded below.

Each cone $\mathcal{K}$ of $\mathcal{A}$ has its collection of chambers, namely those chambers in $\mathcal{C}({\mathcal{A}})$ that lie inside $\mathcal{K}$:

[TABLE]

The poset of interior intersections of the cone $\mathcal{K}$ is the following order ideal within the poset $\mathcal{L}(\mathcal{A})$:

[TABLE]

For each $X$ in $\mathcal{L}_{\mathcal{A}}^{\text{int}}(\mathcal{K})$, its lower interval $[V,X]$ is still a geometric lattice, with same rank function $\operatorname{codim}(X)$, so that one can define the $k^{th}$ (signless) Whitney number of $\mathcal{K}$ by

[TABLE]

along with their generating function ${\sf Poin}(\mathcal{K},t):=\sum_{k=0}^{n}\thinspace c_{k}(\mathcal{K})\thinspace t^{k}$, the Poincaré polynomial for $\mathcal{K}$.

The starting point for our study is the following result of Zaslavsky [32] counting the number $\#\mathcal{C}({\mathcal{K}})$ of chambers of an arrangement $\mathcal{A}$ lying inside one of its cones $\mathcal{K}$.

Theorem 2.9** ([32, Example A, p. 275]).**

Let $\mathcal{K}$ be a cone of an arrangement $\mathcal{A}$ in $V=\mathbb{R}^{n}$. Then

[TABLE]

Zaslavsky proved in his doctoral thesis [31] the better-known special case of Theorem 2.9 for the full arrangement, that is, where $\mathcal{K}=V=\mathbb{R}^{n}$.

The following two examples illustrate Theorem 2.9 for two cones in $A_{3}$.

Example 2.10**.**

Consider the braid arrangement $\mathcal{A}=A_{4}=\{H_{12},H_{13},H_{14},H_{23},H_{24},H_{34}\}$ inside $V=\mathbb{R}^{4}$. On the left below we have drawn a linearly equivalent picture of its intersection with the hyperplane where $x_{1}+x_{2}+x_{3}+x_{4}=0$, isomorphic to $\mathbb{R}^{3}$, and depicted the intersection of the hyperplanes with the unit $2$-sphere in this $3$-dimensional space. Here we pick the cone $\mathcal{K}$ to be the one defined by the halfspace $x_{3}<x_{4}$ for the hyperplane $H_{34}$, and draw the intersection of $H_{34}$ with the unit sphere as the equatorial circle, with the other five hyperplanes $H_{ij}$ depicted as great circles intersecting the hemisphere where $x_{3}<x_{4}$. On the right below the non-hyperplane interior intersection subspaces $X_{\pi}$ are labeled.

H_{14}$$H_{13}$$H_{23}$$H_{12}$$H_{24}$$H_{34}

X_{13|24}$$X_{124}$$X_{14|23}$$X_{123}

Therefore the intersection poset $\mathcal{L}^{\operatorname{int}}(\mathcal{K})$ of this cone is

V=\mathbb{R}^{4}$$H_{12}$$H_{23}$$H_{13}$$H_{24}$$H_{14}$$X_{123}$$X_{124}$$X_{13|24}$$X_{14|23}

We have $(\thinspace c_{0}(\mathcal{K})\thinspace,\thinspace c_{1}(\mathcal{K})\thinspace,\thinspace c_{2}(\mathcal{K})\thinspace)=(1,5,6)$. Summing these gives $1+5+6=12,$ and a quick visual verification assures that there are $12$ chambers in this cone.

Example 2.11**.**

Consider the cone $\mathcal{K}$ of $A_{3}$ in which $x_{3}<x_{4}$ and $x_{1}<x_{2}$. On the left below we have drawn the same picture as Example 2.10 with the cone corresponding to $\mathcal{K}$ shaded. We depict $\mathcal{L}^{\operatorname{int}}(\mathcal{K})$ on the right.

H_{14}$$H_{13}$$H_{23}$$H_{24}$$H_{12}$$H_{34}

V=\mathbb{R}^{4}$$H_{23}$$H_{13}$$H_{24}$$H_{14}$$X_{13|24}

We have $\thinspace c_{0}(\mathcal{K})\thinspace=1$, $\thinspace c_{1}(\mathcal{K})\thinspace=4$, and $\thinspace c_{2}(\mathcal{K})\thinspace=1$. Summing these gives $1+4+1=6=\#\mathcal{C}({\mathcal{K}}).$

For the remainder of this paper, we focus on cones $\mathcal{K}$ inside braid arrangements $A_{n-1}$. It is well-known (see [20, §3.3], for example) and easy to see that such cones correspond bijectively with posets $P$ on $[n]$ via this rule: one has $x_{i}<x_{j}$ for all points in the cone $\mathcal{K}$ if and only if $i<_{P}j$. We will denote the cone associated to $P$ by $\mathcal{K}_{P}$, and abbreviate $\thinspace c_{k}(P)\thinspace:=\thinspace c_{k}(\mathcal{K}_{P})\thinspace$ and ${\sf Poin}(P,t):={\sf Poin}(\mathcal{K}_{P},t)$.

Example 2.12**.**

The cone inside $A_{3}$ in Example 2.10 given by the inequality $x_{3}<x_{4}$ on $V=\mathbb{R}^{4}$ has defining poset $P_{1}$ with order relation $3<_{P_{1}}4$ on $[4]=\{1,2,3,4\}$, while the cone in Example 2.11 given by the inequalities $x_{1}<x_{2}$ and $x_{3}<x_{4}$ has defining poset $P_{2}$ with order relations $1<_{P_{2}}2$ and $3<_{P_{2}}4$. These posets $P_{1},P_{2}$ are shown here:

P_{1}=$$1$$2$$3$$4

P_{2}=$$1$$2$$3$$4

2.2. Preposets, posets, cones and a characterization of $\mathcal{L}^{\operatorname{int}}(\mathcal{K}_{P})$

By Theorem 2.5, the intersection poset $\mathcal{L}(A_{n-1})$ is isomorphic to the set partition lattice $\Pi_{n}$, and hence for each cone $\mathcal{K}_{P}$ in $\mathcal{A}_{n-1}$, one should be able to identify the interior intersection poset $\mathcal{L}^{\operatorname{int}}(\mathcal{K}_{P})$ as some order ideal inside $\Pi_{n}$. This is our next goal, which will be aided by recalling some facts about preposets, posets, binary relations, and cones.

Definition 2.13**.**

Recall that a preposet $Q$ on $[n]$ is a binary relation $Q\subseteq[n]\times[n]$ which is both

• •

reflexive, meaning $(i,i)\in Q$ for all $i$, and

• •

transitive, meaning $(i,j),(j,k)\in Q$ implies $(i,k)\in Q$.

If in addition $Q$ is antisymmetric, meaning $(i,j),(j,i)\in Q$ implies $i=j$, then $Q$ is called a poset on $[n]$; in this case, we sometimes write $i\leq_{Q}j$ when $(i,j)\in Q$.

A set partition $\pi\in\Pi_{n}$ is identified with an equivalence relation $\pi\subseteq[n]\times[n]$ having $(i,j)\in\pi$ when $i,j$ appear in the same block of $\pi$. That is, $\pi$ is reflexive, transitive, and symmetric, meaning $(i,j)\in\pi$ implies $(j,i)\in\pi$. We will sometimes write this binary relation as $i\equiv_{\pi}j$ when $(i,j)\in\pi$.

The union $Q_{1}\cup Q_{2}\subseteq[n]\times[n]$ of two preposets will be reflexive, but possibly not transitive, so not always a preposet. However, the transitive closure operation $Q\mapsto\overline{Q}$ lets one complete it to a preposet $\overline{Q_{1}\cup Q_{2}}$.

We will use a slight rephrasing of the folklore cone-preposet dictionary, as discussed by Postnikov, Reiner, and Williams in [20, Section 3.3]. This dictionary is a bijection between preposets $Q$ on $[n]$ and closed cones of any dimension that are intersections in $V=\mathbb{R}^{n}$ of closed halfspaces of the form $\{x_{i}\leq x_{j}\}$. Under this bijection, any such closed cone $C$ corresponds to a preposet $Q_{C}$ via

[TABLE]

Conversely, any preposet $Q$ on $[n]$ corresponds to a closed cone $C_{Q}$ via

[TABLE]

For a subset $A\subseteq\mathbb{R}^{n}$, denote its interior and relative interior by $\mathsf{int}(A),\mathsf{relint}(A)$. Then for a preposet $Q$,

[TABLE]

Also, one has the following assertions, using the notation of this dictionary:

• •

for $\pi$ in $\Pi_{n}$, the subspace denoted $X_{\pi}$ is the (non-pointed) cone $C_{\pi}$, regarding $\pi$ as a preposet, and

• •

for any poset $P$ on $[n]$, the open $n$-dimensional cone denoted $\mathcal{K}_{P}$ earlier is $\mathsf{relint}(C_{P})$ $(={\sf{int}}(C_{P}))$.

We will need one further dictionary fact.

Proposition 2.14** ([20, Proposition 3.5]).**

For preposets $Q,Q^{\prime}$, one has $C_{Q}\cap C_{Q^{\prime}}=C_{\overline{Q\cup Q^{\prime}}}.$

The following definition will help to characterize the set partitions $\pi$ having $X_{\pi}$ in $\mathcal{L}^{\operatorname{int}}(\mathcal{K}_{P})$.

Definition 2.15**.**

Given a poset $P$ on $[n]$ and a set partition $\pi=\{B_{1},\dots,B_{k}\}$ in $\Pi_{n}$, define a preposet $P/\pi$ on the set $\{B_{1},\dots,B_{k}\}$ as the transitive closure of the (reflexive) binary relation having $(B_{i},B_{j})\in P/\pi$ whenever there exist $p\in B_{i}$ and $q\in B_{j}$ with $p\leq_{P}q$.

Proposition 2.16**.**

For $P$ a poset on $[n]$ and $\pi=\{B_{1},\dots,B_{k}\}$ a set partition in $\Pi_{n}$, the following are equivalent.

1. (i)

$X_{\pi}\in\mathcal{L}^{\operatorname{int}}(\mathcal{K}_{P})$, that is, one has a nonempty intersection $X_{\pi}\cap\mathcal{K}_{P}\neq\varnothing$. 2. (ii)

If $i<_{P}j$, meaning that $i\leq_{P}j$ and $i\neq j$, then $(j,i)\not\in\overline{P\cup\pi}$. 3. (iii)

Every block $B_{i}\in\pi$ is an antichain of $P$, and the preposet $P/\pi$ is actually a poset.

We give a proof of Proposition 2.16 toward the end of this subsection, after some discussion and examples.

Definition 2.17**.**

Let $P$ be a poset on $[n]$. A set partition $\pi$ of $P$ is called a $P$-transverse partition if it satisfies one of the equivalent conditions in Proposition 2.16. We denote by $\Pi^{\pitchfork}(P)$ the induced subset of $\Pi_{n}$ consisting of $P$-transverse partitions.

Remark 2.18*.*

Aguiar and Mahajan [1, p.230] have a similar concept, which they call a prelinear extension of $P$. A prelinear extension of $P$ is equivalent to a $P$-transverse partition $\pi$ together with a linear ordering on the blocks of $\pi$ that extends the partial order $P/\pi$ from Proposition 2.16(iii).

Proposition 2.16 and Corollary 2.6 immediately imply the following corollary.

Corollary 2.19**.**

Let $P$ be a poset on $[n]$. Then $\Pi^{\pitchfork}(P)$ and $\mathcal{L}^{\operatorname{int}}(\mathcal{K}_{P})$ are isomorphic as posets. Consequently,

[TABLE]

Example 2.20**.**

Let $P:=P_{2}$ be the second poset from Example 2.12, with $x_{1}<_{P}x_{2}$ and $x_{3}<_{P}x_{4}$. Then

• •

$\pi=13|24$ is $P$-transverse.

• •

$\pi=12|3|4$ is not $P$-transverse as it fails condition (ii): $1<_{P}2$, but $(2,1)\in\pi\subset\overline{P\cup\pi}.$

• •

$\pi=14|23$ is not $P$-transverse, failing condition (ii): $1<_{P}2$, but $(2,1)\in\overline{P\cup\pi}$, though $(2,1)\not\in P\cup\pi$.

The six $P$-transverse partitions give this subposet $\Pi^{\pitchfork}(P)$ of $\Pi_{4}$ isomorphic to $\mathcal{L}^{\operatorname{int}}(\mathcal{K}_{P})$, as in Example 2.11:

1|2|3|4$$1|23|4$$13|2|4$$1|24|3$$14|2|3$$13|24

It happens that here $|\mu(V,X_{\pi})|=1$ for all $\pi$ in $\Pi^{\pitchfork}(P)$, so that ${\sf Poin}(P,t)=1+4t+t^{2}$.

Example 2.21**.**

Let $P$ be the following poset:

1$$2$$3$$4$$5$$6$$7$$8$$9$$10

Then $\pi=\{\{1,4,7\},\{2,5,10\},\{3,6,8\},\{9\}\}$ in $\Pi_{10}$ is $P$-transverse, represented here by shading the blocks:

Viewed in this way, Proposition 2.16(iii), roughly speaking, states that $\pi$ is $P$-transverse if and only if its blocks are antichains that can be “stacked without crossings” with respect to the Hasse diagram for $P$.

Proof of Proposition 2.16.

We will show a cycle of implications: (i) $\Rightarrow$ (ii) $\Rightarrow$ (iii) $\Rightarrow$ (i). (i) implies (ii):

Assume (i), so that there exists some $\mathbf{x}$ in $\mathbb{R}^{n}$ lying in the nonempty set

[TABLE]

where the first equality comes from the definition of $\mathcal{K}_{P}$ and $C_{P}$, the second from the fact that $\mathcal{K}_{P},C_{P}$ are full $n$-dimensional, the third from Proposition 2.14, and the fourth from equation (11) above. Now to see that (ii) holds, given any pair $i,j$ with $i<_{P}j$, then $x_{i}<x_{j}$ since $\mathbf{x}\in\mathcal{K}_{P}$, but then since $(i,j)\in P\subseteq\overline{P\cup\pi}$, the conditions above imply $(j,i)\not\in\overline{P\cup\pi}$, as desired for (ii).

(ii) implies (iii):

Assume (ii) holds. Then every block $B$ of $\pi$ must be an antichain in $P$, else there exists $i\neq j$ in $B$ with $i<_{P}j$, and then $(j,i)\in\pi\subseteq\overline{P\cup\pi}$, contradicting (ii).

Now suppose for the sake of contradiction that $P/\pi$ is not a poset. Since $P/\pi$ is a preposet, it can only fail to be antisymmetric, that is, there are blocks $B\neq B^{\prime}$ of $\pi$ having both $(B,B^{\prime}),(B^{\prime},B)$ in $P/\pi$. Since both $P,\pi$ are transitive binary relations, this means there must exist a (periodic) sequence of elements of the form

[TABLE]

alternating relations $(p_{i},p_{i+1})$ lying in $P$ and in $\pi$. Then $p_{1}<_{P}p_{2}$ and $(p_{2},p_{1})\in\overline{P\cup\pi}$, contradicting (ii).

(iii) implies (i):

Assume (iii), that is, the blocks of $\pi$ are antichains of $P$, and $P/\pi$ is a poset. One can then reindex the blocks of $\pi$ such that $(B_{1},B_{2},\ldots,B_{k})$ is a linear extension of $P/\pi$. Use this indexing to define a point $\mathbf{x}\in\mathbb{R}^{n}$ whose $p^{th}$ coordinate $x_{p}=i$ if $p$ lies in block $B_{i}$ of $\pi$.

We claim $\mathbf{x}$ lies in $X_{\pi}\cap\mathcal{K}_{P}$, verifying (i). By construction $\mathbf{x}$ lies in $X_{\pi}$, since its coordinates are constant within the blocks of $\pi$. To verify $\mathbf{x}\in\mathcal{K}_{P}$, given $p<_{P}q$, one must check that $x_{p}<x_{q}$. Assume that $p,q$ lie in blocks $B_{i},B_{j}$ of $\pi$, so that $x_{p}=i$ and $x_{q}=j$. Since the blocks of $\pi$ are antichains in $P$ and $p<_{P}q$, one has $i\neq j$, and since $(B_{1},B_{2},\ldots,B_{k})$ is a linear extension of $P/\pi$, one must have $i<j$, that is, $x_{p}<x_{q}$. ∎

It will help later in identifying $P$-transverse partitions to also have the following recursive characterization.

Proposition 2.22**.**

Let $P$ be a poset on $[n]$, and $\pi$ a set partition of $[n]$. Then $\pi$ lies in $\Pi^{\pitchfork}(P)$ if and only if it contains a block $B$ with these two properties:

• (a)

$B$* is a subset of the minimal elements of $P$, and*

• (b)

if $\hat{\pi}:=\pi\backslash\{B\}$ and $\hat{P}$ is the poset on $[n]\backslash B$ obtained from $P$ by removing the elements in $B$, then $\hat{\pi}\in\Pi^{\pitchfork}(\hat{P})$.

Proof.

For the forward implication, assume $\pi\in\Pi^{\pitchfork}(P)$ and $B\in\pi$ is a minimal block. We use Proposition 2.16(iii) to show that (a) and (b) hold. For (a), assume there exists some $x\in B$ which is not minimal in $P$, i.e., there is some $y\in P$ with $x>_{P}y$. The block $B$ is an antichain, so necessarily $y\not\in B$ and so $y$ lies in a block $B^{\prime}\neq B$ of $\pi$. But then $B>_{P/\pi}B^{\prime}$ contradicts the minimality of $B$ in $P/\pi$.

For (b), note that the blocks of $\hat{\pi}$ are a subset of the blocks of $\pi$, so they are antichains in $P$ and necessarily also antichains in $\hat{P}$. Furthermore, the preposet $\hat{P}/\hat{\pi}$ on the blocks of $\hat{\pi}$ must be a poset, else if $B^{\prime},B^{\prime\prime}$ were two blocks of $\hat{\pi}$ having $B^{\prime}\geq_{\hat{P}/\hat{\pi}}B^{\prime\prime}$ and $B^{\prime\prime}\geq_{\hat{P}/\hat{\pi}}B^{\prime}$, then these same two blocks $B^{\prime},B^{\prime\prime}$ in $\pi$ and would have have $B^{\prime}\geq_{P/\pi}B^{\prime\prime}$ and $B^{\prime\prime}\geq_{P/\pi}B^{\prime}$, a contradiction.

For the backward implication, assume $\pi$ in $\Pi_{n}$ has a block $B$ satisfying properties (a), (b). We use Proposition 2.16(i) to show that $\pi\in\Pi^{\pitchfork}(P)$. Since $\hat{\pi}\in\Pi^{\pitchfork}(\hat{P})$, there is a point $\hat{\mathbf{x}}\in\mathbb{R}^{[n]\setminus B}$ in the (nonempty) set $X_{\hat{\pi}}\cap\mathcal{K}_{\hat{P}}$, i.e., the coordinates of $\mathbf{x}$ are constant within each block of $\hat{\pi}$, and $\hat{\mathbf{x}}_{p}<\hat{\mathbf{x}}_{q}$ whenever $p<_{\hat{P}}q$. Let $\mathfrak{m}:=\min_{\mathbb{R}}\{\hat{\mathbf{x}}_{p}:p\in[n]\setminus B\}$ be the smallest coordinate of $\hat{\mathbf{x}}$, and then extend $\hat{\mathbf{x}}\in\mathbb{R}^{[n]\setminus B}$ to a point $\mathbf{x}\in\mathbb{R}^{n}$ by assigning all of the new coordinates $x_{p}$ for $p\in B$ to have the same value, but strictly smaller than $\mathfrak{m}$, e.g., $x_{p}:=\mathfrak{m}-1$ for all $p\in B$. One then checks that this $\mathbf{x}$ lies in $X_{\pi}\cap\mathcal{K}_{P}$: it lies in $\mathcal{K}_{p}$ due to the fact that $B$ is a subset of the minimal elements of $P$, and it lies in $X_{\pi}$ because it is constant on the new block $B$ of $\pi$ not already in $\hat{\pi}$, as well as constant on the blocks of $\hat{\pi}$. ∎

2.3. More examples of $\Pi^{\pitchfork}(P)$

Example 2.23**.**

Given a poset $P$ on $[n]$, its dual or opposite poset $P^{\mathsf{opp}}$ has the same underlying set $[n]$, but with opposite order relation: $i\leq_{P}j$ if and only if $j\leq_{P^{\mathsf{opp}}}i$. One can readily check that conditions (ii) and (iii) in Proposition 2.16 are self-dual in the sense that $\pi$ in $\Pi_{n}$ is $P$-transverse if and only if it is $P^{\mathsf{opp}}$-transverse. Consequently, $\Pi^{\pitchfork}(P^{\mathsf{opp}})=\Pi^{\pitchfork}(P)$, and hence ${\sf Poin}(P^{\mathsf{opp}},t)={\sf Poin}(P,t).$

Example 2.24**.**

Given posets $P_{1},P_{2}$, respectively, their ordinal sum $P_{1}\oplus P_{2}$ is the poset whose underlying set is the disjoint union $P_{1}\sqcup P_{2}$, and having order relations $x\leq_{P_{1}\oplus P_{2}}y$ if either

• •

$x,y\in P_{i}$ and $x_{\leq_{P_{i}}}y$ for some $i=1,2$, or

• •

$x\in P_{1}$ and $y\in P_{2}$.

If the underlying sets for $P_{1},P_{2}$ are $[n_{1}],[n_{2}]$, one can readily check from either of Proposition 2.16 (ii) or (iii) that a partition $\pi$ of $[n_{1}]\sqcup[n_{2}]$ is $P_{1}\oplus P_{2}$-transverse if and only if it is of form $\pi=\{A_{1},\ldots,A_{k},B_{1},\ldots,B_{\ell}\}$ where $\pi_{1}=\{A_{i}\}_{i=1}^{k}$ and $\pi_{2}=\{B_{j}\}_{j=1}^{\ell}$ are $P_{1}$-transverse and $P_{2}$-transverse partitions of $[n_{1}]$ and $[n_{2}]$, respectively. Bearing in mind that $V=\mathbb{R}^{n_{1}+n_{2}}=V_{1}\oplus V_{2}$ where $V_{i}=\mathbb{R}^{n_{i}}$ for $i=1,2$, this gives isomorphisms

[TABLE]

and therefore also

[TABLE]

2.4. Linear extensions, $P$-transverse permutations,

and proof of Theorem 1.1

We recall here the bijection between the chambers of braid arrangement $A_{n-1}$ inside a cone $\mathcal{K}_{P}$ and the linear extensions of $P$. We then define $P$-transverse permutations and use them to combinatorially re-interpret ${\sf Poin}(P,t)$.

Definition 2.25**.**

Given two posets $P,Q$ on $[n]$, say that $Q$ extends $P$ if $i\leq_{P}j$ implies $i\leq_{Q}j$, that is, $P\subseteq Q$ as binary relations on $[n]$, or equivalently, their cones satisfy $\mathcal{K}_{Q}\subseteq\mathcal{K}_{P}$. When $Q$ is a total or linear order $\sigma_{1}<\cdots<\sigma_{n}$ on $[n]$, we identify it with a permutation $\sigma=\sigma_{1}\dots\sigma_{n}$, and call $\sigma$ a linear extension of $P$. Let $\operatorname{LinExt}(P)$ denote the set of all linear extensions of $P$.

Example 2.2 noted that chambers of the braid arrangement $A_{n-1}$ are of the form $\mathcal{K}_{\sigma}$ for permutations $\sigma$. Then $\mathcal{K}_{\sigma}$ is a chamber lying in the cone $\mathcal{C}({\mathcal{K}_{P}})$ if and only if $\sigma$ lies in $\operatorname{LinExt}(P)$, giving a bijection

[TABLE]

See also [27, Example 1.3]. Consequently, as noted in (4) one has

[TABLE]

Example 2.26**.**

The poset $P$ defined by $1<_{P}2$ and $3<_{P}4$ from Example 2.11 has six linear extensions, shown here labeling the chambers in $\mathcal{C}({\mathcal{K}_{P}})$:

3412$$3124$$3142$$1234$$H_{12}$$1324$$1342$$H_{34}

Recall our goal of finding combinatorial interpretations for ${\sf Poin}(P,t)$. Comparing the expression for ${\sf Poin}(P,t)$ given by Corollary 2.19 in terms of $P$-transverse partitions $\pi$, and the interpretation of $\mu(V,X_{\pi})$ in terms of permutations given in (10) motivates the following definition.

Definition 2.27**.**

Given a poset $P$ on $[n]$, a $P$-transverse permutation is a permutation $\sigma$ in $\mathfrak{S}_{n}$ for which the set partition $\pi$ given by the cycles of $\sigma$ is a $P$-transverse partition. Denote by $\mathfrak{S}^{\pitchfork}(P)$ the set of all $P$-transverse permutations.

Corollary 2.19 and equation (10) then immediately imply this interpretation for ${\sf Poin}(P,t)$.

Thereom 1.1. For any poset $P$ on $[n]$, one has

[TABLE]

*In particular, setting $t=1$, one has $\#\operatorname{LinExt}(P)=\#\mathfrak{S}^{\pitchfork}(P)$. *

2.5. An aside on cones in reflection arrangements

We digress here to generalize Theorem 1.1 from posets $P$ parametrizing cones in the type $A_{n-1}$ reflection arrangement, to any real reflection arrangement.

We start first at the level of generality of a complex reflection group $W$ acting on $V=\mathbb{C}^{n}$. This means that $W$ is a finite subgroup of $GL(V)\cong GL_{n}(\mathbb{C})$ generated by (complex, unitary, pseudo-) reflections, which are elements $w$ in $W$ whose fixed space $V^{w}=H$ is a hyperplane, that is, a codimension one $(\mathbb{C}-)$linear subspace. Let $\mathcal{A}_{W}$ denote the arrangement of all such reflecting hyperplanes, and $\mathcal{L}(\mathcal{A}_{W})$ its poset of intersections, as before. This generalizes the type $A_{n-1}$ setting, where $W=\mathfrak{S}_{n}$ and $\mathcal{L}(\mathcal{A}_{W})\cong\Pi_{n}$ is the poset of set partitions of $[n]$. There is also a well-known generalization of the map $\mathfrak{S}_{n}\rightarrow\Pi_{n}$ that sends a permutation $\sigma$ to the set partition $\pi=\{B_{1},B_{2},\ldots,B_{k}\}$ whose blocks $B_{i}$ are the cycles of $\sigma$, given by

[TABLE]

Orlik and Solomon proved [18, §4] the following facts about this map.

Proposition 2.28**.**

For any finite complex reflection group $W$, the map defined in (12) has these properties:

• (a)

The map is well-defined: $V^{w}$ is an intersection of reflecting hyperplanes, so it lies in $\mathcal{L}(\mathcal{A}_{W})$.

• (b)

The map surjects $W\twoheadrightarrow\mathcal{L}(\mathcal{A}_{W})$.

• (c)

The Möbius function values for lower intervals $[V,X]$ in $\mathcal{L}(\mathcal{A}_{W})$ can be expressed via this map as

[TABLE]

Proof.

Parts (a) and (b) are rephrasings of [18, Lemma (4.4)] and part (c) is [18, Lemma (4.7)]. However, we include here a shorter proof, due to C. Athanasiadis444A version of this proof for real reflection groups appears (in Greek) within the proof of Theorem 5.1 on pages 33-34 in the Masters Thesis of Athanasiadis’ student C. Savvidou.. Recall that the values $\mu(V,X)$ are uniquely determined by the identity $\delta_{X,V}=\sum_{Y:V\supseteq Y\supseteq X}\mu(V,Y)$ where $\delta_{X,V}=1$ if $X=V$ and [math] otherwise. It therefore suffices to check these equalities:

[TABLE]

where here $W_{X}$ denotes the subgroup of $W$ that fixes $X$ pointwise. The last equality follows from Steinberg’s Theorem [29, Thm. 1.5]: he showed $W_{X}$ is generated by the reflections whose hyperplane contains $X$, so that $W_{X}=\{1\}$ when $X=V$ (implying $\sum_{w\in W_{X}}\det(w)=\det(1)=1$), and otherwise if $X\neq V$, summing the (nontrivial) character $\det(-)$ over $W_{X}$ yields $\sum_{w\in W_{X}}\det(w)=0$. ∎

For real reflection groups, part (c) above has the following reformulation, generalizing equation (10) above; see [1, Lemma 5.17], [17, §2, pp. 413-414], and [22, Prop. 7.2].

Corollary 2.29**.**

Let $W$ be a finite real reflection group acting on $V=\mathbb{R}^{n}$. For any $X$ in $\mathcal{L}(\mathcal{A}_{W})$, one has

[TABLE]

Proof.

Note $W$ acts orthogonally. We claim that any $w$ acting orthogonally on $\mathbb{R}^{n}$ has $\det(w)=(-1)^{n-\dim(V^{w})}$; given this, Lemma 2.28(c) would finish the proof. To see this claim, note that the eigenvalues $\lambda$ of $w$ lie on the unit circle in $\mathbb{C}$ , so $\lambda\overline{\lambda}=1$. If the eigenvalue $\lambda$ has multiplicity $m_{\lambda}$, then $m_{\overline{\lambda}}=m_{\lambda}$. Thus

[TABLE]

Modulo two, however, we have

[TABLE]

and so $\det(w)=(-1)^{n-\dim(V^{w})}$. ∎

We specialize now to real reflection groups $W$. Here it is known (see, e.g., [14, Chap. 1]) that $W$ permutes the chambers $\mathcal{C}({\mathcal{A}_{W}})$ simply transitively. Thus by fixing a choice of base chamber $C_{0}$, every other chamber $wC_{0}$ has a unique label by some $w$ in $W$, giving a bijection $\mathcal{C}({\mathcal{A}_{W}})\leftrightarrow W$.

Cones $\mathcal{K}$ inside the reflection arrangement $\mathcal{A}_{W}$ correspond to what were called parsets by the third author [21, Chap. 3], or Coxeter cones by Stembridge [30], where they were studied as well-behaved generalizations of posets $P$ on $[n]$. In particular, the set of chambers $\mathcal{C}({\mathcal{K}})$ inside $\mathcal{K}$ generalizes the set $\operatorname{LinExt}(P)$ of linear extensions of $P$. For a cone $\mathcal{K}$ in $\mathcal{A}_{W}$, we consider as before the subposet $\mathcal{L}^{\operatorname{int}}(\mathcal{K})$ of intersection subspaces interior to $\mathcal{K}$, playing the role of the $P$-transverse set partitions $\Pi^{\pitchfork}(P)$ in type $A_{n-1}$. Generalizing the $P$-transverse permutations $\mathfrak{S}^{\pitchfork}(P)$, define the subset

[TABLE]

The real reflection group generalization of Theorem 1.1 is the following.

Theorem 2.30**.**

Any cone $\mathcal{K}$ in the reflection arrangement $\mathcal{A}_{W}$ for a finite real reflection group $W$ has

[TABLE]

In particular, setting $t=1$, one has $\#\mathcal{C}({\mathcal{K}})=\#W^{\pitchfork}(\mathcal{K})$.

Proof.

[TABLE]

where the second equality used Corollary 2.29, and the last equality used the definition of $W^{\pitchfork}(\mathcal{K})$. ∎

3. Bijecting $P$-transverse permutations and linear extensions:

proof of Theorem 1.2

The goal in the next few subsections is to define mutually inverse bijections

[TABLE]

along with the notion of $P$-left-to-right maxima for $\sigma$ in $\operatorname{LinExt}(P)$, to prove this result from the Introduction.

Theorem 1.2. For any poset $P$ on $[n]$, one has a bijection $\Phi:\mathfrak{S}^{\pitchfork}(P)\to\operatorname{LinExt}(P)$ sending the number of cycles to the number of $P$-left-to-right-maxima. Therefore,

[TABLE]

*where ${\sf LRmax}_{P}(\sigma)$ denotes the number of $P$-left-to-right maxima of $\sigma$. *

To this end, we first recall the special case the antichain poset $P={\sf Antichain}_{n}$ on $[n]$, where $\Phi$ is known as the fundamental bijection [28, Proposition 1.3.1]. A permutation $\tau$ in $\mathfrak{S}_{n}$ sending $i\mapsto\tau(i)$ may be written

• •

in a one-line notation as $\tau=[\tau(1),\tau(2),\ldots,\tau(n)]$, or

• •

in a two-line notation $\tau=\begin{pmatrix}1&2&\cdots&n\\ \tau(1)&\tau(2)&\cdots&\tau(n)\end{pmatrix}$, or

• •

in various cycle notations that list the $\tau$-orbits on $[n]$, called its cycles, in some arbitrarily chosen order, with each cycle listed as $(j,\tau(j),\tau^{2}(j),\ldots)$ for some arbitrary choice of the first element $j$.

One way to make the choices non-arbitrary and put the cycle notation in standard form insists that the first element $j$ listed within each cycle $\tau^{(i)}$ is the maximum element of the cycle, and then insists that the cycles $\tau^{(1)},\tau^{(2)},\dots$ are listed with their maximum elements in increasing order as integers, that is, $j_{1}<_{\mathbb{Z}}j_{2}<_{\mathbb{Z}}\cdots$. The fundamental bijection $\mathfrak{S}_{n}\rightarrow\mathfrak{S}_{n}$ sends $\tau$ to $\Phi(\tau):=\sigma=[\sigma_{1},\sigma_{2},\ldots,\sigma_{n}]$ by erasing the parentheses around the standard form cycle notation for $\tau$.

Example 3.1**.**

The permutation $\tau=[7,5,9,4,2,8,3,6,1]$ in $\mathfrak{S}_{9}$ in one-line notation can also be written in two-line notation and factored according to its $\tau$-orbits or cycles

[TABLE]

Its cycle notation in standard form and image $\sigma=\Phi(\tau)$ are then

[TABLE]

The inverse map $\Psi$ starts with $\sigma=[\sigma_{1},\ldots,\sigma_{n}]$ in one-line notation, and determines where to re-insert the parenthesis pairs in the sequence to obtain the standard form for the cycles of $\tau$. One only needs to know the locations of the left parentheses, since then the right parenthesis locations are determined. There will be one left parenthesis just to the left of each $\sigma_{j}$ which is a left-to-right maximum (or LR-maximum for short) in $\sigma$, meaning $\sigma_{j}=\max_{\mathbb{Z}}\{\sigma_{1},\sigma_{2},\dots,\sigma_{j}\}$. It is not hard to check that $\Phi,\Psi$ are mutual inverses, and if $\sigma=\Phi(\tau)$, one has $\mathsf{cyc}(\tau)={\sf LRmax}(\sigma)$, the number of LR-maxima of $\sigma$.

3.1. The map $\Phi:\mathfrak{S}^{\pitchfork}(P)\to\operatorname{LinExt}(P)$

To define the map $\Phi:\mathfrak{S}^{\pitchfork}(P)\to\operatorname{LinExt}(P)$ on a $P$-transverse permutation $\tau$, we will first use the $P$-transverse partition $\pi$ whose blocks are the cycles of $\tau$ to separate the blocks of $\pi$ and the elements of $P$ into levels, and then define a notion of essential elements. Recall that because $\sigma$ lies in $\mathfrak{S}^{\pitchfork}(P)$, meaning $\pi$ lies in $\Pi^{\pitchfork}(P)$, the quotient preposet $P/\pi$ on the blocks of $\pi$ is actually a poset. This leads to the following definition:

Definition 3.2**.**

Say that a block of $\pi$ which is minimal in $P/\pi$ is of Level $1$. For $k\geq 2$, the blocks of $\pi$ of Level $k$ are the minimal ones in the poset obtained from $P/\pi$ by removing all blocks of Level less than $k$.

In other words, a block $B$ of $\pi$ is of Level $k$ if and only if

[TABLE]

or even more concretely, $k$ is the maximum among integers $\ell$ with the property that there exist blocks $B=:B_{1},B_{2},\dots,B_{\ell}$ of $\pi$ and elements $x_{i}>_{P}y_{i+1}$ with $x_{i}\in B_{i},y_{i+1}\in B_{i+1}$ for each $i=1,2,\ldots,\ell-1$.

For an element $x$ in $[n]$, define the Level of $x$ to be the Level of the unique block of $\pi$ containing $x$.

Definition 3.3**.**

An element $x$ is essential if $x$ has Level $k$ and there exists some $y$ of Level $k-1$ with $x>_{P}y$; by convention, all Level $1$ elements $x$ are essential.

In order to define the map $\Phi:\mathfrak{S}^{\pitchfork}(P)\to\operatorname{LinExt}(P)$, we first introduce a standard form for a $P$-transverse permutation $\tau$. Let $\tau$ have cycle partition $\pi=\{B_{1},\ldots,B_{m}\}$, lying in $\Pi^{\pitchfork}(P)$. List the cycles of $\tau$ in order $\tau^{(1)},\tau^{(2)},\ldots,\tau^{(m)}$, with the block $B_{i}$ of $\pi$ corresponding to the cycle $\tau^{(i)}$, and write the cycle $\tau^{(i)}$ as $\tau^{(i)}=(x_{i},\tau(x_{i}),\tau^{2}(x_{i}),\dots)$ for some $x_{i}$ in $B_{i}$. Then this listing is the standard form of $\tau$ if and only if it has the following properties:

• •

If blocks $B_{i},B_{j}$ have Levels $k,k+1$ in $\pi$, respectively, then their indices satisfy $i\leq_{\mathbb{Z}}j$.

• •

For each $i$, the first element $x_{i}$ listed in the cycle $\tau^{(i)}=(x_{i},\tau(x_{i}),\tau^{2}(x_{i}),\dots)$ is the maximal essential element of $B_{i}$; by Lemma 3.5(b), below, every block $B_{i}$ contains an essential element.

• •

If $B_{i},B_{j}$ are blocks of Level $k$ with $i<_{\mathbb{Z}}j$ then $x_{i}<_{\mathbb{Z}}x_{j}$.

Following the fundamental bijection, the map $\Phi:\mathfrak{S}^{\pitchfork}(P)\to\operatorname{LinExt}(P)$ is defined by putting $\tau\in\mathfrak{S}^{\pitchfork}(P)$ into standard form and erasing parenthesis. The following example illustrates this process.

Example 3.4**.**

Let $P$ be the following poset on $[13]$:

13$$1$$9$$11$$6$$7$$2$$5$$3$$10$$4$$12$$8

Let

[TABLE]

and the poset $P/\pi$, drawn as a poset on the cycles of $\tau$, looks as follows:

(13,1)$$(9)$$(11,7)$$(12,5,8,2)$$(6,3)$$(10)$$(4)

One can check that

• •

the Level $1$ cycles are $(13,1),(9),(4)$, with essential elements $1,4,9,13$,

• •

there is one Level $2$ cycle $(11,7)$, with one essential element $7$,

• •

the Level $3$ cycles are $(6,3),(12,5,8,2)$, with essential elements $2,3,5$,

• •

there is one Level $4$ cycle $(10)$, with one essential element $10$.

Here is $\tau$ in standard form, with essential elements overlined, and Levels separated by bars:

[TABLE]

Removing the parentheses (and bars), one obtains its image under $\Phi$:

[TABLE]

The next lemma is used to prove the image of $\Phi$ lies in $\operatorname{LinExt}(P)$, and $\mathfrak{S}^{\pitchfork}(P)\overset{\Phi}{\rightarrow}\operatorname{LinExt}(P)$ is bijective.

Lemma 3.5**.**

Let $P$ be a poset on $[n]$ and $\tau\in\mathfrak{S}^{\pitchfork}(P)$. Then the following properties hold:

1. (a)

For each $k\geq 1$, the Level $k$ elements of $[n]$ form an antichain in $P$. 2. (b)

Every cycle contains at least one essential element. 3. (c)

The image $\Phi(\tau)$ of $\tau$ is a linear extension of $P$.

Proof.

For (a), assume that there were two comparable elements $x<_{P}y$ with $x,y$ both of Level $k$. Either $x,y$ lie in the same block of $\pi$, contradicting $P$-transversality, or they lie in different blocks of $\pi$, which would be comparable in $P/\pi$, contradicting both blocks being of Level $k$.

For (b), note that our most concrete description of a block $B$ having Level $k$ shows that there exist blocks $B=:B_{1},B_{2},\dots,B_{k}$ of $\pi$ and elements $x_{i}>_{P}y_{i+1}$ with $x_{i}\in B_{i},y_{i+1}\in B_{i+1}$ for each $i=1,2,\ldots,k-1$. But then this shows that the block $B_{2}$ must be of Level $k-1$, and $x_{1}>_{P}y_{2}$ shows $x_{1}$ is essential in $B_{1}(=B)$.

For (c), one can show via induction on $k$ that the restriction of $\Phi(\tau)$ to the order ideal of elements of $P$ having Level at most $k$ is a linear extension. In both the base case $k=1$, and in the inductive step, one notes that one can add in the elements of Level $k$ in any order, because they form an antichain by part (a). ∎

3.2. The inverse map $\Psi$

To define the inverse map $\Psi=\Phi^{-1}$ on a linear extension $\sigma$, we proceed similarly to the previous subsection. We first cut $\sigma$ into consecutive strings, suggestively called Levels, define a notion of essential elements of $\sigma$, and a notion of $P$-left-to-right-maximum.

Definition 3.6**.**

Given $\sigma=[\sigma_{1},\ldots,\sigma_{n}]$ in $\operatorname{LinExt}(P)$, we recursively break $\sigma$ into disjoint contiguous sequences $[\sigma_{i},\sigma_{i+1},\ldots,\sigma_{i+j}]$, each forming an antichain of $P$, and each maximal in the sense that the slightly longer sequence $[\sigma_{i},\sigma_{i+1},\ldots,\sigma_{i+j},\sigma_{i+j+1}]$ is not an antichain of $P$:

• •

Let $[\sigma_{1},\sigma_{2},\dots,\sigma_{r}]$ be the longest initial segment of $\sigma$ whose underlying set $\{\sigma_{1},\sigma_{2},\dots,\sigma_{r}\}$ is an antichain of $P$; call $\{\sigma_{1},\sigma_{2},\dots,\sigma_{r}\}$ the Level 1 elements of $\sigma$.

• •

The Level 2 elements of $\sigma$ are $\{\sigma_{r+1},\sigma_{r+2},\ldots,\sigma_{s}\}$, where $[\sigma_{r+1},\sigma_{r+2},\ldots,\sigma_{s}]$ is the longest initial segment of $[\sigma_{r+1},\sigma_{r+2},\ldots,\sigma_{n}]$ that forms an antichain in $P$.

• •

Similarly, for $k\geq 3$, the Level $k$ elements of $\sigma$ are defined as follows: if the union of all elements of Levels $1,2,\ldots,k-1$ are $\{\sigma_{1},\sigma_{2},\dots,\sigma_{t}\}$, then the set of Level $k$ elements is the underlying set of the longest initial segment in $[\sigma_{t+1},\sigma_{t+2},\ldots,\sigma_{n}]$ that forms an antichain in $P$.

Definition 3.7**.**

As in the previous section, say $x$ in $\sigma$ is essential if $x$ has Level $k$ in $\sigma$ and there exists an element $y$ of Level $k-1$ in $\sigma$ with $x>_{P}y$; again by convention, Level $1$ elements of $\sigma$ are all essential.

Definition 3.8**.**

Say that an element $x$ is a $P$-left-to-right-maximum of $\sigma$, or $P$-LR-maximum for short, if $x$ is essential in $\sigma$, and $x$ appears as a LR-maximum in the usual sense among the subsequence of essential elements of $\sigma$ having the same Level as $x$. In other words, if $x$ has Level $k$, it is a $P$-LR-maximum if the subsequence of essential Level $k$ elements is $[\sigma_{i_{1}},\sigma_{i_{2}},\ldots,\sigma_{i_{r}}]$ for some indices $i_{1}<i_{2}<\cdots<i_{r}$, and there is some $j$ with $1\leq j\leq r$ for which $\sigma_{i_{j}}=x=\max\{\sigma_{i_{1}},\sigma_{i_{2}},\ldots,\sigma_{i_{j}}\}$. We denote the number of $P$-LR maxima of $\sigma$ by ${\sf LRmax}_{P}(\sigma)$.

A map $\Psi:\operatorname{LinExt}(P)\to\mathfrak{S}_{n}$ can now be defined in much the same way as for the fundamental bijection: starting with $\sigma=[\sigma_{1},\ldots,\sigma_{n}]$ in $\operatorname{LinExt}(P)$, one must determine where to re-insert the parenthesis pairs in the sequence to recover the cycles of $\tau$. In fact, one only needs to know the locations of the left parentheses, since then the right parenthesis locations are determined, and there will be one left parenthesis just to the left of each $x=\sigma_{j}$ which is a $P$-LR maximum.

Example 3.9**.**

Let $P$ be the poset in Example 3.4 and let

[TABLE]

which is $\Phi(\tau)$ from Example 3.4. The Level decomposition of $\sigma$, with essential elements overlined, looks like

[TABLE]

Now create cycles by placing left parentheses just before each $P$-LR maximum:

[TABLE]

The resulting cycle structure gives the $P$-transverse permutation $\Psi(\sigma)$:

[TABLE]

which is the $P$-transverse permutation $\tau$ in Example 3.4.

Note that it is not yet clear that the image of $\Psi$ lies in $\mathfrak{S}^{\pitchfork}(P)$, but it will follow from the proof of Theorem 1.2. First we need a technical lemma.

Lemma 3.10**.**

Let $\sigma\in\operatorname{LinExt}(P)$ and $\tau\in\mathfrak{S}^{\pitchfork}(P)$ such that $\sigma=\Phi(\tau)$. Then the set of Level $k$ elements of $\tau$ is precisely the set of Level $k$ elements of $\sigma$.

Proof.

We prove this by induction on $k$. For the base case ($k=1$), note that the Level $1$ elements of $\tau$ will form an initial segment $\{\sigma_{1},\sigma_{2},\ldots,\sigma_{\ell}\}$ of $\sigma=[\sigma_{1},\ldots,\sigma_{n}]$, by the definition of $\Phi$. These Level $1$ elements of $\tau$ will also form an antichain of $P$ by Lemma 3.5(a). On the other hand, we claim that the longer initial segment $\{\sigma_{1},\sigma_{2},\ldots,\sigma_{\ell},\sigma_{\ell+1}\}$ cannot form an antichain in $P$, because $\sigma_{\ell+1}$ is of Level $2$ in $\tau$ by definition of $\Phi$, and it is also essential in $\tau$ because it is leftmost in its cycle in the standard form for $\tau$, and all such elements are essential. Thus $\sigma_{\ell+1}>_{P}\sigma_{i}$ for some $i=1,2,\ldots,\ell$, showing that the longer segment is not an antichain of $P$. By definition of $\Psi$, this means that the Level $1$ elements of $\sigma$ will be those in the shorter segment $\{\sigma_{1},\sigma_{2},\ldots,\sigma_{\ell}\}$.

For the inductive step ($k\geq 2$), we perform the same argument as in the base case, but replace $P,\tau,\sigma$ with their counterparts $\hat{P},\hat{\tau},\hat{\sigma}$ in which the elements of Level $1,2,\ldots,k-1$ have been removed. One must check that $\hat{\tau}$ lies in $\mathfrak{S}^{\pitchfork}(\hat{P})$, but this is straightforward, because if $\sigma,\hat{\sigma}$ have cycle partitions $\pi,\hat{\pi}$, then $\hat{P}/\hat{\pi}$ is obtained from $P/\pi$ by removing its minimal blocks. ∎

Proof of Theorem 1.2.

Note $\#\mathfrak{S}^{\pitchfork}(P)=\#\operatorname{LinExt}(P)$ by Theorem 1.1, and $\Phi$ maps $\mathfrak{S}^{\pitchfork}(P)\to\operatorname{LinExt}(P)$ by Lemma 3.5(c). We therefore claim that it suffices to check $\Psi(\Phi(\tau))=\tau$ for all $\tau$ in $\mathfrak{S}^{\pitchfork}(P)$. This will imply that $\Phi$ is injective, hence bijective, with $\Psi$ its inverse bijection, and thus the image of $\Psi$ is $\mathfrak{S}^{\pitchfork}(P)$. Note that by construction, if $\Psi(\sigma)=\tau$, then ${\sf LRmax}_{P}(\sigma)=\mathsf{cyc}(\tau)$, so Theorem 1.2 would follow.

By Lemma 3.10, we have if $\sigma=\Phi(\tau)$, then the sets of Level $k$ elements of $\sigma$ and $\tau$ coincide. It follows immediately from Lemma 3.10 that the essential elements in $\sigma$ and $\tau$ coincide, since in each case, their definition uses only the order $P$ and the partition by Levels. Therefore, we can focus our attention on each Level $k$ separately, where the definition of $\Phi$ and $\Psi$ coincides almost exactly with their definition in the fundamental bijection, ignoring the non-essential elements carried along in each Level. It then follows that $\Psi(\Phi(\tau))=\tau$ via the same argument as for the fundamental bijection. ∎

4. Disjoint Unions of Chains and proofs of Theorems 1.3 and 1.4

The goal of this section is to understand ${\sf Poin}(P,t)$ for a poset $P$ that is a disjoint union of chains. Although Theorem 1.2 applies to any poset, when $P$ is a disjoint union of chains, there turns out to be another elegant expression for ${\sf Poin}(P,t)$ stemming from Foata’s theory of multiset permutations, generalizing equation (8) for the antichain poset $P$.

In Subsection 4.1 we review Foata’s theory of multiset permutations, in particular his work with the intercalation product and prime cycle decompositions. Subsection 4.2 reviews its relation to partial commutation monoids. Subsection 4.3 shows how the results in Section 3 can be rephrased in terms of multiset permutations when $P$ is a disjoint union of chains. Theorem 1.3 is also proved in this subsection. Finally Subsection 4.4 employs Foata’s theory to give a generalization of MacMahon’s Master Theorem which specializes to Theorem 1.4, a generating function compiling the Poincaré polynomials for disjoint unions of chains.

4.1. Multiset Permutations

This subsection gives background on the theory of multiset permutations as introduced by Foata in his PhD thesis [9, Section 3.2], and extended in later publications [10, Chapters 3-5]. It also appears in Knuth [15, Section 5.1.2].

Definition 4.1**.**

Recall that a (weak) composition $\overline{a}=(a_{1},\ldots,a_{\ell})$ of $n$ is a sequence of nonnegative integers having sum $|a|:=\sum_{i}a_{i}=n$. We will regard $\overline{a}$ as specifying the multiplicities in a multiset ${M(\overline{a})}:=\{1^{a_{1}},2^{a_{2}},\dots,\ell^{a_{\ell}}\}$, that is, a set with repetitions

[TABLE]

A multiset permutation $\sigma=[\sigma_{1},\ldots,\sigma_{n}]$ is a rearrangement of the elements of ${M(\overline{a})}$, which we will often write in a two-line notation that generalizes that of permutations:

[TABLE]

We denote the set of all multiset permutations of ${M(\overline{a})}$ by $\mathfrak{S}_{M(\overline{a})}$. For any $\sigma\in\mathfrak{S}_{M(\overline{a})}$, we call ${M(\overline{a})}$ the support of $\sigma$, and write ${M(\overline{a})}=\mathsf{supp}(\sigma)$.

Example 4.2**.**

The composition $\overline{a}=(2,3,2,3)$ gives the multiplicities of the multiset

[TABLE]

Then the following multiset permutation $\sigma$ is an element of $\mathfrak{S}_{M(\overline{a})}$:

[TABLE]

Foata [9, §3.2] defined an associative intercalation product operation on multiset permutations $(\sigma,\rho)\mapsto\sigma\intercal\rho$. Knuth [15, §5.1.2] describes it algorithmically: think of $\sigma,\rho$ in two-line notation as sequences of columns $\left(\begin{smallmatrix}i\\ j\end{smallmatrix}\right)$, and juxtapose these sequences of columns. Then perform swaps to sort the columns according to their top entries, never swapping two with the same top entry. For example,

[TABLE]

Definition 4.3**.**

For each $\ell$, the intercalation monoid $\operatorname{Int}_{\ell}$ is the submonoid of all multiset permutations $\sigma$ whose support $M=\{1^{a_{1}},2^{a_{2}},\dots,\ell^{a_{\ell}}\}$ involves only the letters in $\{1,2,\ldots,\ell\}$. The empty permutation $()$ is the identity element for $\intercal$, since $()\intercal\sigma=\sigma=\sigma\intercal()$.

Note that, just as permutations in the symmetric group $\mathfrak{S}_{n}$ do not commute in general, the monoid $\operatorname{Int}_{\ell}$ is not commutative. For example

[TABLE]

However, one can check that $\sigma\intercal\rho=\rho\intercal\sigma$ when $\sigma,\rho$ are disjoint, that is, $\mathsf{supp}(\sigma)\cap\mathsf{supp}(\rho)=\emptyset$.

Definition 4.4**.**

Say $\sigma$ in $\operatorname{Int}_{\ell}$ is prime if the only factorizations $\sigma=\rho\intercal\tau$ have either $\rho=()$ or $\tau=()$.

Example 4.5**.**

The permutation $\begin{pmatrix}2&4&5&7\\ 5&7&4&2\end{pmatrix}$ is prime. However, $\begin{pmatrix}1&1&2&3\\ 2&3&1&1\end{pmatrix}$ is not prime, since

[TABLE]

On the other hand $\begin{pmatrix}2&4&5&7\\ 5&7&2&4\end{pmatrix}$ is not prime, even though its support is multiplicity free, since

[TABLE]

It is not obvious, but turns out to be true that $\sigma$ is prime if and only if both

• •

$\mathsf{supp}(\sigma)=M$ is multiplicity free, that is, $M$ is a set not a multiset, and

• •

$\sigma$ has only one cycle when considered as an ordinary permutation of the set $M$.

We therefore call prime elements prime cycles. More generally, one has the following.

Theorem 4.6** (Foata, 1969 [10, 15]).**

Let $\sigma$ be a multiset permutation. Then $\sigma$ has a decomposition into a product of prime cycles. That is, there exist $k\geq 0$ prime cycles $\sigma^{(1)},\dots,\sigma^{(k)}$ such that

[TABLE]

Further, this cycle decomposition of $\sigma$ is unique up to successively interchanging pairs of adjacent prime cycles with disjoint support. In particular $k$ is unique.

Definition 4.7**.**

Call $\mathsf{pcyc}(\sigma):=k$ the number of prime cycles in the decomposition of $\sigma$ from Theorem 4.6.

Example 4.8**.**

The element $\sigma$ from Example 4.2 has $\mathsf{pcyc}(\sigma)=4$ and two prime cycle decompositions

[TABLE]

We describe here an algorithm to find a prime cycle decomposition of a multiset permutation, which can be deduced from [15, §5.1.2], and illustrate how it produces the first of the two decompositions in Example 4.8. Encode a multiset permutation $\sigma$ in $\mathfrak{S}_{M(\overline{a})}$ as two pieces of data:

• •

a directed graph $D_{\sigma}$ on vertex set $\{1,2,\ldots,\ell\}$ having one copy of the directed arc $i\to j$ for each occurrence of the column $\binom{i}{j}$ in its two-line notation, along with

• •

specification for each vertex $x$ in $\{1,2,\ldots,\ell\}$ the linear ordering of the arcs $x\to y$ emanating from $x$, indicating the left-to-right ordering of the corresponding columns in the two-line notation.

The resulting digraphs are those with the outdegree equal to the indegree equal to $a_{i}$ for each $i$. E.g., the $\sigma$ from Example 4.8 has this directed graph $D_{\sigma}$, with linear orderings indicated on the arcs out of each vertex:

[TABLE]

With this identification, one factors $\sigma$ recursively. First produce a prime cycle $\sigma^{(1)}$ for which

[TABLE]

via the following algorithm that takes a directed walk in $D_{\sigma}$.

• •

Start at the smallest vertex $i_{0}$ in $\{1,2,\ldots,\ell\}$ with outdegree $a_{i_{0}}\geq 1$, and follow its first outward arc $i_{0}\to i_{1}$. Then follow $i_{1}$’s first outward arc $i_{1}\to i_{2}$, follow $i_{2}$’s first outward arc $i_{2}\to i_{3}$, etc.

• •

Repeat until first arriving at a previously-visited555During the process, when one enters a new vertex $i_{j}$ along an arc $i_{j_{1}}\to i_{j}$, there will always be at least one outward arc $i_{j}\to i_{j+1}$ leaving $i_{j}$, because each vertex started with its indegree matching its outdegree. vertex $i_{s}$, say $i_{s}=i_{r}$ with $r<s$; possibly $r=s-1$.

• •

The directed circuit $C$ of arcs $i_{r}\to i_{r+1}\to i_{r+2}\to\cdots\to i_{s-1}\to i_{s}(=i_{r})$ corresponds to a prime cycle $\sigma^{(1)}$ that one can factor out to the left as in (14): by construction, each of its corresponding columns $\binom{i_{t}}{j_{t}}$ occurs as the leftmost column of $\sigma$ having $i_{t}$ as its top element.

• •

Complete the factorization recursively, replacing $\sigma$ by $\hat{\sigma}$, removing the arcs $C$ from $D_{\sigma}$ to give $D_{\hat{\sigma}}$.

Example 4.9**.**

Here is the algorithm for $\sigma$ above, with dotted arrows showing the directed walks in $D_{\sigma}$:

[TABLE]

[TABLE]

4.2. Partial Commutation Monoids

It will be helpful to view the intercalation monoid $\operatorname{Int}_{\ell}$ as a partial commutation monoid. We briefly review some relevant facts about partial commutation monoids.

Definition 4.10**.**

Given a set $\mathbb{A}$, which we call an alphabet and a subset of its pairs $C\subseteq\binom{\mathbb{A}}{2}$, the associated partial commutation monoid $\mathcal{M}$ is defined to be the set of equivalence classes on words $\alpha_{1}\alpha_{2}\dots\alpha_{k}$ in the alphabet $\mathbb{A}$ under the equivalence relation

[TABLE]

if $\{\alpha_{i},\alpha_{i+1}\}\in C$.

From this perspective, Foata’s Theorem 4.6 asserts that $\operatorname{Int}_{\ell}$ is a partial commutation monoid, whose associated alphabet $\mathbb{A}$ is the set of all prime cycles, and $C$ the pairs of prime cycles with disjoint supports.

For later use, we point out the following (nontrivial) proposition, see [15, §5.1.2, Exercise 11] and [28, Exercise 3.123]. Given a factorization of an element $\alpha=\alpha_{1}\alpha_{2}\dots\alpha_{k}$ in $\mathcal{M}$ a partial commutation monoid, define a poset $\mathcal{P}_{\alpha}$ on $[k]$ as the transitive closure of the binary relation containing $(i,j)\in\mathcal{P}_{\alpha}$ when $i<_{\mathbb{Z}}j$ and either $\alpha_{i}=\alpha_{j}$ or $\alpha_{i}\alpha_{j}\not\equiv\alpha_{j}\alpha_{i}$.

Proposition 4.11**.**

Given a factorization of $\alpha=\alpha_{1}\alpha_{2}\dots\alpha_{k}$ in $\mathcal{M}$ a partial commutation monoid,

1. (1)

$\mathcal{P}_{\alpha}$* does not depend on the choice of factorization of $\alpha$, and* 2. (2)

there is a bijection between $\operatorname{LinExt}(\mathcal{P}_{\alpha})$ and the factorizations of $\alpha$ given by

[TABLE]

Example 4.12**.**

The multiset permutation $\sigma$ from Example 4.8 had two prime cycle factorizations

[TABLE]

corresponding to the two linear extensions of the poset $\mathcal{P}_{\sigma}$ on $[4]$ with this Hasse diagram:

1$$2$$3$$4

4.3. Connection with linear extensions and $P$-transverse permutations

We wish to use Foata’s prime cycle decomposition to define a bijection $\operatorname{LinExt}(P_{\overline{a}})\rightarrow\mathfrak{S}^{\pitchfork}(P_{\overline{a}})$, and use this to prove Theorem 1.3.

We begin with an easy identification of $\operatorname{LinExt}(P_{\overline{a}})$ with $\mathfrak{S}_{M(\overline{a})}$. For this purpose, given any weak composition $\overline{a}=(a_{1},\ldots,a_{\ell})$ of $n$, consider two labelings of $P_{\overline{a}}$, one by the elements $\{1,2,\ldots,n\}$ which we will call the standardized labeling, and the second by the elements of the multiset ${M(\overline{a})}$, which we will call the multiset labeling. The standardized labeling labels the first chain $\mathsf{a}_{1}$ by $1,2,\ldots,a_{1}$ from bottom-to-top, then the second chain $\mathsf{a}_{2}$ by $a_{1}+1,\ldots,a_{1}+a_{2}$ from bottom-to-top, and so on. The multiset labeling labels the elements in the first chain $\mathsf{a}_{1}$ all by $1$, the second chain $\mathsf{a}_{2}$ all by $2$, etc.

Example 4.13**.**

For $n=10$ and $\overline{a}=(2,3,2,3)$, the standardized and multiset labelings of $P_{\overline{a}}$ are

1$$2$$3$$4$$5$$6$$7$$8$$9$$10

1$$1$$2$$2$$2$$3$$3$$4$$4$$4

With this in hand, the following proposition is a straightforward observation.

Proposition 4.14**.**

For any weak composition $\overline{a}$ of $n$, one has a bijection

[TABLE]

replacing $\lambda_{i}$ by its corresponding multiset label $\sigma_{i}$, that is, if $\lambda_{i}$ lies on the $j^{th}$ chain $\mathsf{a}_{j}$ in $P_{\overline{a}}$, then $\sigma_{i}:=j$.

Proof.

The inverse map recovers $\lambda$ from $\sigma$ by labeling the $a_{j}$ occurrences of the value $j$ within $\sigma$ from left-to-right with the integers in the interval $[a_{1}+a_{2}+\cdots+a_{j-1}+1,a_{1}+a_{2}+\cdots+a_{j-1}+a_{j}]$. ∎

Example 4.15**.**

For $\overline{a}=(2,3,2,3)$, this bijection maps $\lambda=[3,8,9,6,1,4,2,7,10,5]$ in $\operatorname{LinExt}(P_{\overline{a}})$ to

[TABLE]

We can now define a map $\varphi:\operatorname{LinExt}(P_{\overline{a}})\rightarrow\mathfrak{S}_{n}$, which will turn out to be a bijection onto $\mathfrak{S}^{\pitchfork}(P_{\overline{a}})$.

Definition 4.16**.**

Fix a weak composition $\overline{a}$ of $n$. Given $\lambda$ in $\operatorname{LinExt}(P_{\overline{a}})$,

• •

let $\sigma\in\mathfrak{S}_{M(\overline{a})}$ be its corresponding multiset permutation from Proposition 4.14,

• •

label the entries in the top line of $\sigma$’s two-line notation with subscripts $1,2,\ldots,n$ from left-to-right,

• •

use Foata’s Theorem 4.6 to decompose $\sigma=\sigma^{(1)}\intercal\cdots\intercal\sigma^{(\ell)}$ into prime cycles $\sigma^{(i)}$, carrying along the subscripts in the top line, and finally

• •

replace each prime cycle $\sigma^{(i)}$ with the cyclic permutation $\tau^{(i)}$ of the subscripts of its top line.

Then $\varphi(\lambda):=\tau=\tau^{(1)}\cdots\tau^{(\ell)}$ in $\mathfrak{S}_{n}$.

Example 4.17**.**

We continue Example 4.15. Let $\overline{a}=(2,3,2,3)$ and $\lambda=[3,8,9,6,1,4,2,7,10,5]$ in $\operatorname{LinExt}(P_{\overline{a}})$. Subscript the top line of its corresponding $\sigma$ in $\mathfrak{S}_{M(\overline{a})}$, and factor as in Theorem 4.6, carrying along subscripts:

[TABLE]

Replacing each prime cycle $\sigma^{(i)}$ with the cycle $\tau^{(i)}$ on its subscripts gives $\varphi(\lambda)=\tau\in\mathfrak{S}_{10}$:

[TABLE]

We can now prove Theorem 1.3, whose statement we recall here. Theorem 1.3. For any composition $\overline{a}$ of $n$, the disjoint union $P_{\overline{a}}$ of chains has a bijection

[TABLE]

with $\mathsf{cyc}(\tau)=\mathsf{pcyc}(\sigma)$, the number of prime cycles in Foata’s unique decomposition for $\sigma$. Thus

[TABLE]

Proof.

We claim that the above map $\varphi:\operatorname{LinExt}(P_{\overline{a}})\rightarrow\mathfrak{S}_{n}$ is the desired bijection. Since Theorem 1.1 showed $\#\operatorname{LinExt}(P_{\overline{a}})=\#\mathfrak{S}^{\pitchfork}(P_{\overline{a}})$, it suffices to show that the image of $\varphi$ lies in $\mathfrak{S}^{\pitchfork}(P_{\overline{a}})$, and that $\varphi$ is injective.

To see that every $\lambda$ in $\operatorname{LinExt}(P_{\overline{a}})$ has $\varphi(\lambda)=\tau$ lying in $\mathfrak{S}^{\pitchfork}(P_{\overline{a}})$, we will induct on the number $\ell$ of cycles in $\tau=\tau^{(1)}\cdots\tau^{(\ell)}$, which is also the number of prime cycles in the decomposition $\sigma=\sigma^{(1)}\intercal\cdots\intercal\sigma^{(\ell)}$. By definition of $\mathfrak{S}^{\pitchfork}(P_{\overline{a}})$, we must check that the cycle partition $\pi=\{B_{1},\ldots,B_{\ell}\}$ of $\tau$ lies in $\Pi^{\pitchfork}(P_{\overline{a}})$, where $B_{i}$ is the set underlying the cycle $\tau^{(i)}$. To this end, assume that we produce the factorization $\sigma=\sigma^{(1)}\intercal\cdots\intercal\sigma^{(\ell)}$ according to the algorithm presented in Subsection 4.1, and let us check that the block $B=B_{1}$ underlying $\tau^{(1)}$ satisfies the two properties (a), (b) in the recursive characterization of $\Pi^{\pitchfork}(P_{\overline{a}})$ from Proposition 2.22:

• •

For (a), the elements of $B$ are all minimal in $P_{\overline{a}}$ because, in the initial factorization step $\sigma=\sigma^{(1)}\intercal\hat{\sigma},$ each column $\binom{i}{j}$ in the two-line notation of $\sigma^{(1)}$ is the leftmost column of $\sigma$ having $i$ as its top element, so it corresponds to the bottom element in the $i^{th}$ chain $\mathsf{a_{i}}$ of $P_{\overline{a}}$.

• •

For (b), note that after that initial factorization step, the poset $\hat{P_{\overline{a}}}$ and partition $\hat{\pi}=\{B_{2},\ldots,B_{\ell}\}$ will correspond to $\hat{\sigma}$ in the above factorization, coming from a $\hat{\lambda}$ in $\operatorname{LinExt}(\hat{P_{\overline{a}}})$ with $\varphi(\hat{\lambda})=\hat{\sigma}$ for which the result holds by induction on $\ell$.

To show that $\varphi$ is injective, we must give an algorithm to recover $\lambda$ from $\tau=\varphi(\lambda)$. It would be equivalent to recover $\tau$’s multiset labeled image $\sigma$ in $\mathfrak{S}_{M(\overline{a})}$ from the bijection in Proposition 4.14. Factoring $\tau$ lets us recover its unordered set of cycles $\{\tau^{(1)},\ldots,\tau^{(\ell)}\}$, and hence also the unordered set of prime cycles $\{\sigma^{(1)},\ldots,\sigma^{(\ell)}\}$ that will appear in an intercalation factorization of $\sigma$. We would like to know how to properly index $\{\sigma^{(i)}\}_{i=1,\ldots,\ell}$, up to interchanging commuting elements, so that we could recover $\sigma$ as their intercalation product $\sigma=\sigma^{(1)}\intercal\cdots\intercal\sigma^{(\ell)}$. We claim that this (partial) ordering is already contained in the information of the unordered set of cycles $\{\tau^{(i)}\}_{i=1,\ldots,\ell}$ as follows. When two prime cycles $\sigma^{(r)},\sigma^{(s)}$ do not commute, it is because they share a common element $i$, so there must exist two elements $x,y$ in $[n]$ that come from the $i^{th}$ chain $\mathsf{a}_{i}$ in the standardized labeling of $P_{\overline{a}}$, with $x\in\tau^{(r)},y\in\tau^{(s)}$. If $x<_{\mathbb{Z}}y$, then $\sigma^{(r)}$ must occur to the left of $\sigma^{(s)}$ in the intercalation product. ∎

4.4. Proof of Theorem 1.4

Our goal here is to find a generating function compiling the Poincaré polynomials ${\sf Poin}(P_{\overline{a}},t)$ for all compositions $\overline{a}$ of length $\ell$. This uses more of Foata’s theory for the intercalation monoid $\operatorname{Int}_{\ell}$, similar to his deduction of MacMahon’s Master Theorem.

Since each multiset permutation $\sigma$ has only finitely many intercalation factorizations $\sigma=\rho\intercal\tau$, one can define a convolution algebra on the set of functions $\phi:\operatorname{Int}_{\ell}\rightarrow\mathbb{Z}$ with pointwise addition:

[TABLE]

Let $\zeta:\operatorname{Int}_{\ell}\rightarrow\mathbb{Z}$ denote the zeta function defined by $\zeta(\sigma)=1$ for all $\sigma$ in $\operatorname{Int}_{\ell}$. The zeta function has a unique convolutional inverse $\mu$, called the Möbius function. Foata proved that the Möbius function can be expressed by the following explicit formula

[TABLE]

where $\sigma\in\operatorname{Int}_{\ell}$ is simple if all the letters of $\sigma$ are distinct, that is, $\mathsf{supp}(\sigma)$ is a set, not a multiset. This may be formulated as an identity in a completion $\mathbb{Z}[[\operatorname{Int}_{\ell}]]:=\left\{\sum_{\sigma\in\operatorname{Int}_{\ell}}z_{\sigma}\sigma:z_{\sigma}\in\mathbb{Z}\right\}$ of the monoid algebra $\mathbb{Z}[\operatorname{Int}_{\ell}]$, allowing infinite $\mathbb{Z}$-linear combinations of elements of $\operatorname{Int}_{\ell}$ (see [10, Théorème 2.4]):

[TABLE]

Now introduce an $\ell\times\ell$ matrix $B:=(b_{ij})_{i,j=1,2,\ldots,\ell}$ of indeterminates, and let $\mathbb{Z}[[b_{ij},t]]$ be the (usual, commutative) power series ring in $\{b_{ij}\}_{i,j=1}^{\ell}$ along with one further indeterminate $t$. One can then define a ring homomorphism

[TABLE]

where if $\sigma=\begin{pmatrix}i_{1}&i_{2}&\cdots&i_{n}\\ \sigma_{1}&\sigma_{2}&\cdots&\sigma_{n}\end{pmatrix}$ then $\underline{b}_{\sigma}:=\prod_{k=1}^{n}b_{i_{k}\sigma_{k}}$.

Applying the homomorphism $u_{t}$ to both sides of (16) gives a $t$-version of MacMahon’s Master Theorem.

Theorem 4.18**.**

In $\mathbb{Z}[[b_{ij},t]]$ one has the identity

[TABLE]

Remark 4.19*.*

Setting $t=1$ in Theorem 4.18 gives an identity in $\mathbb{Z}[[b_{ij}]]$:

[TABLE]

which is equivalent to an identity in Foata’s proof of the (commutative) MacMahon Master Theorem, as we recall here. Introduce two sets of $\ell$ variables $\mathbf{x}=(x_{1},\ldots,x_{\ell}),\mathbf{y}=(y_{1},\ldots,y_{\ell})$ related by the matrix $B$ of indeterminates as follows: $\mathbf{y}=B\mathbf{x}$, that is, $y_{i}=\sum_{j}b_{ij}x_{j}$. Then MacMahon’s Master Theorem is this identity in $\mathbb{Z}[[b_{ij}]]$:

[TABLE]

where $\mathbf{x}^{\overline{a}}:=x_{1}^{a_{1}}\cdots x_{\ell}^{a_{\ell}}.$ It is not hard to check that the left sides and right sides of (18) and (17) are the same: the left side of (17) needs to be grouped according to the multiplicity vector $\overline{a}$ giving the support $\mathsf{supp}(\sigma)$, and the right side must be reinterpreted in terms of the permutation expansion of a determinant.

Remark 4.20*.*

Theorem 4.18 is similar in spirit to Garoufalidis-Lê-Zeilberger’s quantum MacMahon Master Theorem [11, Theorem 1] (see also Konvalinka-Pak [16, Theorem 1.2]). Their quantum version inserts a $(-q)^{-\text{inv}(\sigma)}$ in order to produce a $q$-determinant, but $\text{inv}(\sigma)\not=\mathsf{pcyc}(\sigma)$.

We now specialize $b_{ij}=x_{j}$ in Theorem 4.18 to deduce Theorem 1.4, whose statement we recall here.

Theorem 1.4. * For $\ell=1,2,\ldots$, one has*

[TABLE]

*where $e_{j}(\mathbf{x}):=\sum_{1\leq i_{1}<\cdots<i_{j}\leq\ell}x_{i_{1}}\cdots x_{i_{j}}$ is the $j^{th}$ elementary symmetric function. *

Proof.

Setting $b_{ij}=x_{j}$ in Theorem 4.18 gives

[TABLE]

Let us manipulate both sides of equation (19). On the left side, grouping terms according to $\mathsf{supp}(\sigma)$ gives

[TABLE]

On the right side of (19), any subset $H\subseteq[\ell]$ of cardinality $j\geq 1$ satisfies

[TABLE]

by (8). Therefore grouping according to $j=\#H$, and noting $\sum_{\begin{subarray}{c}H\subseteq[\ell]:\\ \#H=j\end{subarray}}\prod_{k\in H}x_{k}=e_{j}(\mathbf{x})$ lets one rewrite the sum inside the parentheses on the right side of (19) as this:

[TABLE]

So far this gives

[TABLE]

Now perform two more substitutions: first replace $t$ by $t^{-1}$, giving this

[TABLE]

and then replace $x_{i}$ by $tx_{i}$ for $i=1,2,\ldots,\ell$, so that $\mathbf{x}^{\overline{a}}\mapsto t^{|\overline{a}|}\mathbf{x}^{\overline{a}}$ and $e_{j}(\mathbf{x})\mapsto t^{j}e_{j}(\mathbf{x})$, giving this

[TABLE]

Comparison of the left side with Theorem 1.3 shows that this last equation is Theorem 1.4. ∎

Remark 4.21*.*

We justify here the claim from the Introduction that Theorem 1.4 generalizes the formula (8):

[TABLE]

Since ${\sf Antichain}_{\ell}=P_{\overline{a}}$ where $\overline{a}=(1,1,\ldots,1)$, we seek to explain why the coefficient of $x_{1}\dots x_{\ell}$ in the power series on the right side in Theorem 1.4 should be $1(1+t)(1+2t)\cdots(1+(\ell-1)t)$. Introducing the abbreviation $\langle t\rangle_{j}:=(t-1)(2t-1)\cdots((j-1)t-1)$, the right side in Theorem 1.4 can be rewritten and expanded as

[TABLE]

If we let $A_{\ell}$ denote the coefficient of $x_{1}\cdots x_{\ell}$ in this series, then it suffices to explain why

[TABLE]

For this coefficient extraction, it is safe to replace each $e_{j}(\mathbf{x})=e_{j}(x_{1},x_{2},\ldots,x_{\ell})$ in (20) with an infinite variable version $e_{j}(\mathbf{x})=e_{j}(x_{1},x_{2},\ldots)$. Extracting the coefficient of $x_{1}\cdots x_{\ell}$ on the right side in (20) shows

[TABLE]

To explain (21), note that each ordered set partition $\hat{\pi}$ of $[\ell+1]$ can be obtained from a unique ordered set partition $\pi=(B_{1},\dots,B_{n})$ of $[\ell]$ as follows: either $\hat{\pi}$ has added $\ell+1$ into one of the preexisting blocks $B_{i}$ of $\pi$, or $\hat{\pi}$ has a singleton block $\{\ell+1\}$, inserted into one of the $n+1$ locations in the sequence $(B_{1},\ldots,B_{n})$. Thus having fixed an ordered set partition $\pi$ of $[\ell]$, the sum of $w(\hat{\pi})$ over $\hat{\pi}$ which correspond to $\pi$ is this sum:

[TABLE]

Summing this over all possible $\pi$ gives (21).

5. Posets of width two and proof of Theorem 1.5

The width of a poset $P$ is the maximum size of an antichain in $P$. A famous result of Dilworth from 1950 (see [28, Ch 3, Exer 77(d)]), asserts that the width $d$ of $P$ is the minimum number of chains required in a chain decomposition $P=P_{1}\cup P_{2}\cup\cdots\cup P_{d}$, that is, where each $P_{i}$ is a totally ordered subset $P_{i}\subseteq P$.

Consequently, a poset $P$ of width two can be decomposed into two chains $P=P_{1}\cup P_{2}$, possibly with some order relations between elements of $P_{1}$ and $P_{2}$.

Recall that in the Introduction we defined a descent-like statistic on $\sigma=[\sigma_{1},\ldots,\sigma_{n}]$ in $\operatorname{LinExt}(P)$, as the cardinality $\mathsf{des}_{P_{1},P_{2}}(\sigma):=\#\mathsf{Des}_{P_{1},P_{2}}(\sigma)$ of this set,

[TABLE]

in order to state the following result. Theorem 1.5. For a width two poset decomposed into two chains as $P=P_{1}\cup P_{2}$, one has

[TABLE]

To prove this, we start with the following observation.

Corollary 5.1**.**

For posets $P$ of width two, one has

[TABLE]

where ${\sf pairs}(\pi)$ is the number of two-element blocks $B_{i}$ in $\pi$. In particular, setting $t=1$,

[TABLE]

Proof.

Antichains in $P$ have at most two elements, so Proposition 2.16(iii) implies that $P$-transverse permutations have only $1$-cycles and $2$-cycles. But then this implies that the map $\mathfrak{S}^{\pitchfork}(P)\rightarrow\Pi^{\pitchfork}(P)$ sending a $P$-transverse permutation $\tau$ to the set partition $\pi$ given by its cycles is a bijection, with ${\sf pairs}(\pi)=n-\mathsf{cyc}(\tau)$. The result then follows from Theorem 1.1. ∎

Example 5.2**.**

Let $P=\mathsf{a}\sqcup\mathsf{b}$ be a poset which is a disjoint union of two chains $\mathsf{a},\mathsf{b}$ having $a,b$ elements respectively. One can check that a $P$-transverse partition having ${\sf pairs}(\pi)=k$ is completely determined by the choice of a $k$ element subset $x_{1}<_{P}\cdots<_{P}x_{k}$ from $\mathsf{a}$ and a $k$ element subset $y_{1}<_{P}\cdots<_{P}y_{k}$ from $\mathsf{b}$ to constitute the two-element blocks, as follows: $\{x_{1},y_{1}\},\ldots,\{x_{k},y_{k}\}$. This implies

[TABLE]

This is consistent with $\#\operatorname{LinExt}(\mathsf{a}\sqcup\mathsf{b})=\binom{a+b}{a}$, since setting $t=1$ in the equation above gives

[TABLE]

which is an instance of the Chu-Vandermonde summation.

In light of Corollary 5.1, to prove Theorem 1.5, one would need a bijection from $\operatorname{LinExt}(P)$ to $\Pi^{\pitchfork}(P)$ (or $\mathfrak{S}^{\pitchfork}(P)$) that sends the statistic $\mathsf{des}_{(P_{1},P_{2})}(-)$ to the number of pairs or number of $2$-cycles. Unfortunately, there does not seem to be a consistent labeling of a width two poset $P=P_{1}\cup P_{2}$ to make the bijection $\Psi$ from Section 3 play this role. Nevertheless, having fixed the chain decomposition666So we assume here that $P_{1}\cap P_{2}=\varnothing$, but there may be order relations between elements of $P_{1}$ and $P_{2}$. $P=P_{1}\sqcup P_{2}$, we provide in the proof below such a bijection $\Omega:\operatorname{LinExt}(P)\rightarrow\Pi^{\pitchfork}(P).$

Proof of Theorem 1.5..

We describe $\Omega$ and $\Omega^{-1}$ recursively, via induction on $n:=\#P$. There are two cases, based on whether $P$ has one or two minimal elements.

Case 1. There is a unique minimum element $p_{0}\in P$.

In this case, given $\sigma=[\sigma_{1},\dots,\sigma_{n}]$ in $\operatorname{LinExt}(P)$, we must have $\sigma_{1}=p_{0}$, so that $\{p_{0}\}$ should be a singleton block of $\pi=\Omega(\sigma)$, and one produces the remaining blocks of $\pi$ by applying $\Omega$ recursively to $[\sigma_{2},\ldots,\sigma_{n}]$. This is depicted schematically here:

p_{0}$$\vdots$$\vdots$$(P,[\sigma_{1},\sigma_{2},\dots,\sigma_{n}])

p_{0}$$\vdots$$\vdots$$(P,[{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}p_{0}},\sigma_{2}\dots,\sigma_{n}])

\vdots$$\vdots$$(P-\{p_{0}\},[\sigma_{2},\dots,\sigma_{n}])

For the inverse map $\Omega^{-1}$, given a $P$-transverse partition $\pi$, since the blocks of $\pi$ are antichains in $P$, the unique minimum element $p_{0}$ of $P$ must lie in a singleton block $\{p_{0}\}$ in $\pi$. So make $\Omega^{-1}(\pi)=\sigma$ have $\sigma_{1}=p_{0}$, and construct $[\sigma_{2},\ldots,\sigma_{n}]$ by applying $\Omega^{-1}$ recursively to the $(P-\{p_{0}\})$-transverse partition obtained from $\pi$ by removing the block $\{p_{0}\}$.

Case 2. There are two minimal elements of $P$.

Label these two minimal elements $p_{1},p_{2}$ of $P$ so that $p_{i}\in P_{i}$ for $i=1,2$. Note that this implies that every $\sigma=[\sigma_{1},\sigma_{2},\ldots,\sigma_{n}]$ in $\operatorname{LinExt}(P)$ has either $\sigma_{1}=p_{1}$ or $\sigma_{1}=p_{2}$. Note also that any $P$-transverse partition $\pi$ only has blocks of cardinality $1$ or $2$, which yields two subcases for defining $\Omega$ and $\Omega^{-1}$:

• •

The Subcase 2a for

• –

defining $\Omega$ occurs when $\sigma_{1}=p_{1}$,

• –

defining $\Omega^{-1}$ occurs when $\{p_{1}\}$ appears as a singleton block within $\pi$.

• •

The Subcase 2b for

• –

defining $\Omega$ occurs when $\sigma_{1}=p_{2}$,

• –

defining $\Omega^{-1}$ occurs when $p_{1}$ appears in a two-element block within $\pi$.

Subcase 2a.

When defining $\Omega$, if $\sigma_{1}=p_{1}$, then make $\{p_{1}\}$ a singleton block of $\pi=\Omega(\sigma)$, and produce the remaining blocks of $\pi$ by applying $\Omega$ recursively to $[\sigma_{2},\ldots,\sigma_{n}]$.

p_{1}=\sigma_{1}$$p_{2}\neq\sigma_{1}$$\vdots$$\vdots$$(P,(\sigma_{1},\sigma_{2},\dots,\sigma_{n}))

p_{2}\neq\sigma_{1}$$\vdots$$\vdots$$(P-\{p_{1}\},(\sigma_{2},\dots,\sigma_{n}))

To define $\Omega^{-1}$, if $\{p_{1}\}$ is a singleton block of $\pi$, make $\Omega^{-1}(\pi)=\sigma$ have $\sigma_{1}=p_{1}$, and construct $[\sigma_{2},\ldots,\sigma_{n}]$ by applying $\Omega^{-1}$ recursively to the $(P-\{p_{1}\})$-transverse partition obtained from $\pi$ by removing the block $\{p_{1}\}$.

Subcase 2b.

When defining $\Omega$, if $\sigma_{1}=p_{2}$, then $p_{1}$ appears elsewhere in $\sigma$, say $p_{1}=\sigma_{i+1}$ where $i\geq 1$. Because $\sigma$ lies in $\operatorname{LinExt}(P)$ and $\sigma_{i+1}=p_{1}$ is the minimum element of $P_{1}$, this forces $\sigma_{1},\sigma_{2},\ldots,\sigma_{i}$ to all be elements of $P_{2}$. In this case, add to $\pi$ the singleton blocks $\{\sigma_{1}\},\{\sigma_{2}\},\dots,\{\sigma_{i-1}\}$ along with the two-element block $\{\sigma_{i},\sigma_{i+1}\}=\{\sigma_{i},p_{1}\}$, and compute the rest of $\Omega(\sigma)=\pi$ recursively by replacing $(P,\sigma)$ with $(P-\{\sigma_{1},\sigma_{2}\dots,\sigma_{i+1}\},[\sigma_{i+2},\sigma_{i+3},\dots,\sigma_{n}]).$ Here is the schematic picture:

p_{1}=\sigma_{i+1}$$p_{2}=\sigma_{1}$$\vdots$$\sigma_{i}$$\vdots$$\vdots$$(P,(\sigma_{1},\sigma_{2},\dots,\sigma_{n}))

\vdots$$\vdots$$(P-\{\sigma_{1},\dots,\sigma_{i}\},(\sigma_{i+1},\sigma_{i+2},\dots,\sigma_{n}))

When defining $\Omega^{-1}(\pi)$, if $p_{1}$ appears in some two-element block of $\pi$, then it appears in some block $\{p_{1},p_{2}^{\prime}\}$ for some $p_{2}^{\prime}$ in $P_{2}$. We claim that $\pi$ being $P$-transverse then forces any elements $p<_{P}p_{2}^{\prime}$ in $P_{2}$ to lie in singleton blocks $\{p\}$ of $\pi$. To see this claim, assume not, so that some such $p$ lies in a two-element block of $\pi$, necessarily of the form $\{p_{1}^{\prime},p\}$ for some $p_{1}^{\prime}$ in $P_{1}$ with $p_{1}<_{P}p_{1}^{\prime}$. This leads to a contradiction of Proposition 2.16(ii), since $(p_{1}^{\prime},p_{1})$ would then be a relation in $\overline{P\cup\pi}$ via this transitive chain of relations: $p_{1}^{\prime}\equiv_{\pi}p<_{P}p_{2}^{\prime}\equiv_{\pi}p_{1}.$

In this subcase, list the totally ordered (and possibly empty) collection of all elements $p$ in $P_{2}$ with $p<_{P}p_{2}^{\prime}$ at the beginning of $\sigma$ as $\sigma_{1},\sigma_{2},\ldots,\sigma_{i-1}$, followed by $\sigma_{i}\sigma_{i+1}=p_{2}^{\prime}p_{1}$. Then compute the rest of $\Omega^{-1}(\pi)=\sigma$ recursively, by applying $\Omega^{-1}$ to the $(P-\{\sigma_{1},\sigma_{2},\ldots,\sigma_{i+1}\})$-transverse partition obtained from $\pi$ by removing the singleton blocks $\{\sigma_{1}\},\{\sigma_{2}\},\ldots,\{\sigma_{i-1}\}$ and the two-element block $\{\sigma_{i},\sigma_{i+1}\}=\{p_{2}^{\prime},p_{1}\}$.

It is not hard to check that the two maps $\Omega,\Omega^{-1}$ defined recursively in this way are actually mutually inverse bijections. By construction, $\Omega$ has the property that the two-element blocks of $\pi=\Omega(\sigma)$ are exactly those containing $P$-incomparable pairs $\{\sigma_{i},\sigma_{i+1}\}$ for which $\sigma_{i}\in P_{1}$ and $\sigma_{i+1}\in P_{2}$, as claimed. ∎

Recall from the Introduction that the number of usual descents of a permuation $\sigma$ is defined as $\mathsf{des}(\sigma)=\#\mathsf{Des}(\sigma)$ where

[TABLE]

This was used to define the $P$-Eulerian polynomial in equation (9) as $\sum_{\sigma\in\operatorname{LinExt}(P)}t^{\mathsf{des}(\sigma)}$, assuming that $P$ is naturally labeled, that is, $\operatorname{LinExt}(P)$ contains the identity permutation $\sigma=[1,2,\ldots,n]$.

Corollary 5.3**.**

When $P$ is a width two poset having a chain decomposition $P_{1}\cup P_{2}$ with $P_{1}$ an order ideal of $P$, then the Poincaré polynomial for $P$ coincides with the $P$-Eulerian polynomial:

[TABLE]

Proof.

Let $\#P_{i}=n_{i}$ for $i=1,2$, so that $n=\#P=n_{1}+n_{2}$. One can then choose a natural labeling for $P$ by $[n]$ such that the elements of the order ideal $P_{1}$ are labeled by the initial segment $[n_{1}]=\{1,2,\ldots,n_{1}\}$, and $P_{2}$ is labeled by $\{n_{1}+1,n_{1}+2,\ldots,n\}$. We claim that with this natural labeling, one has $\mathsf{Des}_{(P_{1},P_{2})}(w)=\mathsf{Des}(w)$. This is because the labeling renders one of the conditions in the definition (22) of $\mathsf{Des}_{(P_{1},P_{2})}(\sigma)$ superfluous: assuming that $\sigma_{i}\in P_{2}$ and $\sigma_{i+1}\in P_{1}$, then $\sigma_{i},\sigma_{i+1}$ must already be incomparable in $P$, because otherwise $\sigma_{i}<_{P}\sigma_{i+1}$ (since $\sigma$ was a linear extension of $P$) and then $P_{1}$ being an order ideal would force $\sigma_{i}\in P_{1}$, a contradiction. Now since $P_{1},P_{2}$ are totally ordered in $P$, and $\sigma$ lies in $\operatorname{LinExt}(P)$, one has $\sigma_{i}\in P_{2}$ and $\sigma_{i+1}$ in $P_{1}$ if and only if $\sigma_{i}>_{\mathbb{Z}}\sigma_{i+1}$. That is $\mathsf{Des}_{(P_{1},P_{2})}(w)=\mathsf{Des}(w)$. ∎

Example 5.4**.**

An interesting family of posets to which Corollary 5.3 applies are the posets $P(\lambda/\mu)$ associated with two-row skew Ferrers diagrams $\lambda/\mu$. A Ferrers diagram associated to a partition (of a number) $\lambda=(\lambda_{1},\ldots,\lambda_{\ell})$ has $\lambda_{i}$ square cells drawn left-justified in row $i$. A skew Ferrers diagram $\lambda/\mu$ for two partitions $\lambda,\mu$ having $\lambda_{i}\geq\mu_{i}$ is the diagram for $\lambda$ with the cells occupied by the diagram for $\mu$ removed. There is a poset structure $P(\lambda/\mu)$ on the cells of $\lambda/\mu$ in which a cell $(i,j)$ in row $i$ and column $j$ has $(i,j)\leq_{P(\lambda/\mu)}(i^{\prime},j^{\prime})$ if both $i\leq i^{\prime}$ and $j\leq j^{\prime}$.

When $\lambda/\mu$ has only two parts, we will call it a two-row skew Ferrers diagram. Three examples of such $\lambda/\mu$ and their associated $P(\lambda/\mu)$ are shown below.

[TABLE]

The decomposition $P(\lambda/\mu)=P_{1}\cup P_{2}$ where $P_{i}$ correspond to the cells in row $i$ of $\lambda/\mu$ shows that $P(\lambda/\mu)$ has width two, and furthermore $P_{1}$ forms an order ideal. Therefore Corollary 5.3 implies that for any two-row skew Ferrers diagram $\lambda/\mu$ one has

[TABLE]

On the other hand, there is a well-known bijection between linear extensions $\sigma$ of $P(\lambda/\mu)$ and the standard Young tableaux $Q$ of shape $\lambda/\mu$, which are (bijective) labelings of the cells of the diagram by $[n]$ where $n=\sum_{i}\lambda_{i}-\sum_{i}\mu_{i}$, with the numbers increasing left-to-right in rows and top-to-bottom in columns; see [26, §7.10]. There is also a notion of descent set $\mathsf{Des}(Q)$ for such tableaux, having $i\in\mathsf{Des}(Q)$ whenever $i+1$ labels a cell in a lower row of $Q$ than $i$. However, in general when $\sigma$ corresponds to $Q$, one does not have $\mathsf{des}(\sigma)=\mathsf{des}(Q)$, so that ${\sf Poin}(P(\lambda/\mu),t)$ differs from the generating function $\sum_{Q}t^{\mathsf{des}(Q)}$ of standard tableaux $Q$ shape $\lambda/\mu$ by $\mathsf{des}(Q)$. For example, there are two standard tableaux of shape $\lambda/\mu=(2,1)/(0,0)$

[TABLE]

both having $\mathsf{des}(Q_{i})=1$, however ${\sf Poin}(P(\lambda/\mu),t)=1+t$.

In two special cases, however, they (essentially) coincide.

• •

When $P_{\lambda/\mu}=\mathsf{a}\sqcup\mathsf{b}$ is a disjoint union of two chains, as in Example 5.2, one can check that, if one (naturally) labels $\mathsf{a}\sqcup\mathsf{b}$ so that the elements of the order ideal $\mathsf{b}$ are labeled $1,2,\ldots,b$ while $\mathsf{a}$ is labeled $b+1,b+2,\ldots,b+a$, then one does have $\mathsf{des}(\sigma)=\mathsf{des}(Q)$, and hence

[TABLE]

• •

When $\lambda/\mu$ is a $2\times n$ rectangle, so that $P_{\lambda/\mu}=\mathsf{2}\times\mathsf{n}$ is a Cartesian product poset, then $\sigma$ in $\operatorname{LinExt}(P)$ and standard Young tableaux $Q$ of shape $2\times n$ can both be identified with Dyck paths of semilength $n$, that is, lattice paths from $(0,0)$ to $(2n,0)$ in $\mathbb{Z}^{2}$ taking steps northeast or southeast and staying weakly above the $x$-axis. One can check that

• –

$\mathsf{Des}(\sigma)$ corresponds to valleys (i.e. southwest steps followed by a northeast step), while

• –

$\mathsf{Des}(Q)$ correspond to peaks (i.e. northeast steps followed by a southwest step).

In general, such a Dyck path has one more peak than valley [26, Exercises 6.19(i, ww, aaa)]. Hence one has

[TABLE]

which is the generating function for the Narayana numbers $N(n,k):=\frac{1}{n}\binom{n}{k-1}\binom{n}{k}$ (see [7, p.2] and [26, Exer. 6.36(a)]). Upon setting $t=1$, the Naryana numbers sum to the Catalan number

[TABLE]

Note that for any (non-skew) partition $\lambda$, the celebrated hook-length formula of Frame, Robinson and Thrall [26, Corollary 7.21.6] gives a simple product formula for $\#\operatorname{LinExt}(P(\lambda))=\left[{\sf Poin}(P(\lambda),t)\right]_{t=1}.$

Open Problem 5.5**.**

Combinatorially interpret ${\sf Poin}(P(\lambda),t)$ for other partitions $\lambda$, and in particular, for $m\times n$ rectangular partitions, where $P(\lambda)=\mathsf{m}\times\mathsf{n}$ is a Cartesian product of chains.

Below we give ${\sf Poin}(\mathsf{3}\times\mathsf{n},t)$ for $2\leq n\leq 8$.

[TABLE]

Remark 5.6*.*

Since equation (8) shows that the Poincaré polynomial ${\sf Poin}(P,t)$ for the antichain poset $P=P_{(1^{n})}={\sf Antichain}_{n}$ has only real roots, one might wonder whether this holds for some more general class of posets. It does not hold for all posets, and not even for all disjoint unions of chains $P_{\overline{a}}$, since

[TABLE]

has a pair of non-real complex roots. It can fail even for rectangular Ferrers posets, e.g., $\lambda=(3,3,3)$ has ${\sf Poin}(P(\lambda),t)={\sf Poin}(\mathsf{3}\times\mathsf{3},t)=1+9t+19t^{2}+11t^{3}+2t^{4}$ in the above table, with two non-real complex roots.

On the other hand, computations show that ${\sf Poin}(P,t)$ is real-rooted for all posets $P$ of width two having at most $9$ elements. This leads to the following question.

Question 5.7**.**

Is ${\sf Poin}(P,t)$ real-rooted when the poset $P$ has width two?

Acknowledgements

The authors gratefully acknowledge Dennis Stanton for conversations about MacMahon’s Master Theorem, as well as Anders Björner, Jesus DeLoera, Theo Douvropoulos, Michael Falk, Ira Gessel, Benjamin Steinberg, Volkmar Welker, Chi-Ho Yuen for enlightening discussions and references. They thank Philip Zhang for asking them questions about real-rootedness at the 2019 Mid-Atlantic Algebra, Geometry, and Combinatorics Workshop. Finally, the authors thank an anonymous referee for helpful comments.