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The \(h^*\)-Polynomials of Locally Anti-Blocking Lattice Polytopes and Their \(\gamma \)-Positivity

Abstract

A lattice polytope \(\mathscr {P} \subset \mathbb {R}^d\) is called a locally anti-blocking polytope if for any closed orthant \({\mathbb R}^d_{\varepsilon }\) in \(\mathbb {R}^d\), \(\mathscr {P} \cap \mathbb {R}^d_{\varepsilon }\) is unimodularly equivalent to an anti-blocking polytope by reflections of coordinate hyperplanes. We give a formula for the \(h^*\)-polynomials of locally anti-blocking lattice polytopes. In particular, we discuss the \(\gamma \)-positivity of \(h^*\)-polynomials of locally anti-blocking reflexive polytopes.

Introduction

A lattice polytope is a convex polytope all of whose vertices have integer coordinates. A lattice polytope \({\mathscr {P}}\subset {\mathbb R}_{\ge 0}^d\) of dimension d is called anti-blocking if for any \({\mathbf{y}}=(y_1,\dots ,y_d) \in {\mathscr {P}}\) and \({\mathbf{x}}=(x_1,\dots ,x_d) \in {\mathbb R}^d\) with \(0 \le x_i \le y_i\) for all i, it holds that \({\mathbf{x}}\in {\mathscr {P}}\). Anti-blocking polytopes were introduced and studied by Fulkerson [11, 12] in the context of combinatorial optimization. See, e.g., [35]. For \(\varepsilon \in \{-1,1\}^d\) and \({\mathbf{x}}\in {\mathbb R}^d\), set \(\varepsilon {\mathbf{x}}:=(\varepsilon _1 x_1,\ldots ,\varepsilon _d x_d) \in {\mathbb R}^d\). Given an anti-blocking lattice polytope \({\mathscr {P}}\subset {\mathbb R}_{\ge 0}^d\) of dimension d, we define

$$\begin{aligned} {\mathscr {P}}^{\pm }:=\{\varepsilon {\mathbf{x}}\in {\mathbb R}^d : \varepsilon \in \{-1,1\}^{d}, \, {\mathbf{x}}\in {\mathscr {P}}\}. \end{aligned}$$

Since \({\mathscr {P}}\) is an anti-blocking lattice polytope, \({\mathscr {P}}^\pm \) is convex (and a lattice polytope). Moreover, for any \(\varepsilon \in \{-1,1\}^d\) and \({\mathbf{x}}\in {\mathscr {P}}^{\pm }\), we have \(\varepsilon {\mathbf{x}}\in {\mathscr {P}}^{\pm }\). The polytope \({\mathscr {P}}^{\pm }\) is called an unconditional lattice polytope [23]. In general, \({\mathscr {P}}^{\pm }\) is symmetric with respect to all coordinate hyperplanes. In particular, the origin \(\mathbf{0}\) of \({\mathbb R}^d\) is in the interior \({\text {int}}{\mathscr {P}}^{\pm }\). Given \(\varepsilon =(\varepsilon _1,\ldots , \varepsilon _d) \in \{-1,1\}^d\), let \({\mathbb R}^d_{\varepsilon }\) denote the closed orthant \(\{ (x_1,\ldots , x_d) \in {\mathbb R}^d : x_i \varepsilon _i \ge 0\ \hbox { for all}\ 1 \le i \le d\}\). A lattice polytope \({\mathscr {P}}\subset {\mathbb R}^d\) of dimension d is called locally anti-blocking [23] if, for each \(\varepsilon \in \{-1,1\}^d\), there exists an anti-blocking lattice polytope \({\mathscr {P}}_{\varepsilon } \subset {\mathbb R}_{\ge 0}^d\) of dimension d such that \({\mathscr {P}}\cap {\mathbb R}^d_{\varepsilon }={\mathscr {P}}_{\varepsilon }^{\pm } \cap {\mathbb R}^d_{\varepsilon }\). Unconditional polytopes are locally anti-blocking.

In the present paper, we investigate the \(h^*\)-polynomials of locally anti-blocking lattice polytopes. First, we give a formula for the \(h^*\)-polynomials of locally anti-blocking lattice polytopes in terms of that of unconditional lattice polytopes.

Theorem 1.1

Let \({\mathscr {P}}\subset {\mathbb R}^d\) be a locally anti-blocking lattice polytope of dimension d and for each \(\varepsilon \in \{-1,1\}^d\), let \({\mathscr {P}}_{\varepsilon }\) be an anti-blocking lattice polytope of dimension d such that \({\mathscr {P}}\cap {\mathbb R}^d_\varepsilon = {\mathscr {P}}_{\varepsilon }^\pm \cap {\mathbb R}^d_\varepsilon \). Then the \(h^*\)-polynomial of \({\mathscr {P}}\) satisfies

$$\begin{aligned} h^*({\mathscr {P}}, x)=\frac{1}{2^{d}}\sum _{\varepsilon \in \{-1,1\}^d}\!h^*({\mathscr {P}}_{\varepsilon }^\pm , x). \end{aligned}$$

In particular, \(h^*({\mathscr {P}},x)\) is \(\gamma \)-positive if \(h^*({\mathscr {P}}_{\varepsilon }^{\pm },x)\) is \(\gamma \)-positive for all \(\varepsilon \in \{-1,1\}^d\).

Second, we discuss the \(\gamma \)-positivity of the \(h^*\)-polynomials of locally anti-blocking reflexive polytopes. A lattice polytope is called reflexive if the dual polytope is also a lattice polytope. Many authors have studied reflexive polytopes from viewpoints of combinatorics, commutative algebra, and algebraic geometry. In [15], Hibi characterized reflexive polytopes in terms of their \(h^*\)-polynomials. To be more precise, a lattice polytope of dimension d is (unimodularly equivalent to) a reflexive polytope if and only if the \(h^*\)-polynomial is a palindromic polynomial of degree d. On the other hand, in [23], locally anti-blocking reflexive polytopes were characterized. In fact, a locally anti-blocking lattice polytope \({\mathscr {P}}\subset {\mathbb R}^d\) of dimension d is reflexive if and only if for each \(\varepsilon \in \{-1,1\}^d\), there exists a perfect graph \(G_{\varepsilon }\) on \([d]:=\{1,\ldots ,d\}\) such that \({\mathscr {P}}\cap {\mathbb R}^d_{\varepsilon }={\mathscr {Q}}_{G_{\varepsilon }}^{\pm } \cap {\mathbb R}^d_{\varepsilon }\), where \({\mathscr {Q}}_{G_{\varepsilon }}\) is the stable set polytope of \(G_{\varepsilon }\). Moreover, every locally anti-blocking reflexive polytope possesses a regular unimodular triangulation. This fact and the result of Bruns–Römer [5] imply that its \(h^*\)-polynomial is unimodal.

In the present paper, we discuss whether the \(h^*\)-polynomial of a locally anti-blocking reflexive polytope has a stronger property, which is called \(\gamma \)-positivity. In [31], a class of lattice polytopes \({\mathscr {B}}_G\) arising from finite simple graphs G on [d], which are called symmetric edge polytopes of type B, was introduced. Symmetric edge polytopes of type B are unconditional, and they are reflexive if and only if the underlying graphs are bipartite. Moreover, when they are reflexive, the \(h^*\)-polynomials are always \(\gamma \)-positive. On the other hand, in [30], another family of lattice polytopes \({\mathscr {C}}^{(e)}_P\) arising from finite partially ordered sets P on [d], which are called enriched chain polytopes, was given. Enriched chain polytopes are unconditional and reflexive, and their \(h^*\)-polynomials are always \(\gamma \)-positive. Combining these facts and Theorem 1.1, we know that, for a locally anti-blocking reflexive polytope \({\mathscr {P}}\), if every \({\mathscr {P}}\cap {\mathbb R}_\varepsilon ^d\) is the intersection of \({\mathbb R}^d_{\varepsilon }\) and either an enriched chain polytope or a symmetric edge reflexive polytope of type B, then the \(h^*\)-polynomial of \({\mathscr {P}}\) is \(\gamma \)-positive (Corollary 4.2). By using this result, we show that the \(h^*\)-polynomials of several classes of reflexive polytopes are \(\gamma \)-positive.

In Sect. 5, we will discuss \(\gamma \)-positivity of the \(h^*\)-polynomials of symmetric edge polytopes of type A, which are reflexive polytopes arising from finite simple graphs. In [21], it was shown that the \(h^*\)-polynomials of the symmetric edge polytopes of type A of complete bipartite graphs are \(\gamma \)-positive. We will show that for a large class of finite simple graphs, which includes complete bipartite graphs, the \(h^*\)-polynomials of the symmetric edge polytopes of type A are \(\gamma \)-positive (Sect. 5.1). Moreover, by giving explicit \(h^*\)-polynomials of del Pezzo polytopes and pseudo-del Pezzo polytopes, we will show that the \(h^*\)-polynomial of every pseudo-symmetric simplicial reflexive polytope is \(\gamma \)-positive (Theorem 5.8).

In Sect. 6, we will discuss \(\gamma \)-positivity of \(h^*\)-polynomials of twinned chain polytopes \({\mathscr {C}}_{P,Q} \subset {\mathbb R}^d\), which are reflexive polytopes arising from two finite partially ordered sets P and Q on [d]. In [39], it was shown that twinned chain polytopes \({\mathscr {C}}_{P,Q}\) are locally anti-blocking and each \({\mathscr {C}}_{P,Q} \cap {\mathbb R}_{\varepsilon }^d\) is the intersection of \({\mathbb R}^d_{\varepsilon }\) and an enriched chain polytope. Hence the \(h^*\)-polynomials of \({\mathscr {C}}_{P,Q} \) are \(\gamma \)-positive. We will give a formula for the \(h^*\)-polynomials of twinned chain polytopes in terms of the left peak polynomials of finite partially ordered sets (Theorem 6.3). Moreover, we will define enriched (PQ)-partitions of P and Q, and show that the Ehrhart polynomial of the twined chain polytope \({\mathscr {C}}_{P,Q}\) of P and Q coincides with a counting polynomial of enriched (PQ)-partitions (Theorem 6.8).

This paper is organized as follows: In Sect. 2, we will review the theory of Ehrhart polynomials, \(h^*\)-polynomials, and reflexive polytopes. In Sect. 3, we will introduce several classes of anti-blocking polytopes and unconditional polytopes. In Sect. 4, we will investigate the \(h^*\)-polynomials of locally anti-blocking lattice polytopes. In particular, we will prove Theorem 1.1. We will discuss symmetric edge polytopes of type A in Sect. 5, and twinned chain polytopes in Sect. 6.

Ehrhart Theory and Reflexive Polytopes

In this section, we review the theory of Ehrhart polynomials, \(h^*\)-polynomials, and reflexive polytopes. Let \({\mathscr {P}}\subset {\mathbb R}^d\) be a lattice polytope of dimension d. Given a positive integer m, we define

$$\begin{aligned} L_{{\mathscr {P}}}(m)=|m {\mathscr {P}}\cap {\mathbb Z}^d|. \end{aligned}$$

Ehrhart [10] proved that \(L_{{\mathscr {P}}}(m)\) is a polynomial in m of degree d with the constant term 1. We say that \(L_{{\mathscr {P}}}(m)\) is the Ehrhart polynomial of \({\mathscr {P}}\). The generating function of the lattice point enumerator, i.e., the formal power series

$$\begin{aligned} \text {Ehr}_{\mathscr {P}}(x)=1+\sum \limits _{k=1}^{\infty }L_{{\mathscr {P}}}(k)x^k \end{aligned}$$

is called the Ehrhart series of \({\mathscr {P}}\). It is well known that it can be expressed as a rational function of the form

$$\begin{aligned} \text {Ehr}_{\mathscr {P}}(x)=\frac{h^*({\mathscr {P}},x)}{(1-x)^{d+1}}. \end{aligned}$$

Then \(h^*({\mathscr {P}},x)\) is a polynomial in x of degree at most d with nonnegative integer coefficients [36] and it is called the \(h^*\)-polynomial (or the \(\delta \)-polynomial) of \({\mathscr {P}}\). Moreover, one has \({\text {Vol}}({\mathscr {P}})=h^*({\mathscr {P}},1)\), where \({\text {Vol}}({\mathscr {P}})\) is the normalized volume of \({\mathscr {P}}\).

A lattice polytope \({\mathscr {P}}\subset {\mathbb R}^d\) of dimension d is called reflexive if the origin of \({\mathbb R}^d\) is a unique lattice point belonging to the interior of \({\mathscr {P}}\) and its dual polytope

$$\begin{aligned}{\mathscr {P}}^\vee :=\{{\mathbf{y}}\in {\mathbb R}^d : \langle {\mathbf{x}},{\mathbf{y}}\rangle \le 1 \ \text {for all}\ {\mathbf{x}}\in {\mathscr {P}}\}\end{aligned}$$

is also a lattice polytope, where \(\langle {\mathbf{x}},{\mathbf{y}}\rangle \) is the usual inner product of \({\mathbb R}^d\). It is known that reflexive polytopes correspond to Gorenstein toric Fano varieties, and they are related to mirror symmetry (see, e.g., [3, 7]). In each dimension there exist only finitely many reflexive polytopes up to unimodular equivalence [25] and all of them are known up to dimension 4 [24]. In [15], Hibi characterized reflexive polytopes in terms of their \(h^*\)-polynomials. We recall that a polynomial \(f \in {\mathbb R}[x]\) of degree d is said to be palindromic if \(f(x)=x^df(x^{-1})\). Note that if a lattice polytope of dimension d has interior lattice points, then the degree of its \(h^*\)-polynomial is equal to d.

Proposition 2.1

[15]Let \({\mathscr {P}}\subset {\mathbb R}^d\) be a lattice polytope of dimension d with \(\mathbf{0} \in {\text {int}}{\mathscr {P}}\). Then \({\mathscr {P}}\) is reflexive if and only if \(h^*({\mathscr {P}},x)\) is a palindromic polynomial of degree d.

Next, we review some properties of polynomials. Let \(f= \sum _{i=0}^{d}a_i x^i\) be a polynomial with real coefficients and \(a_d \ne 0\). We now focus on the following properties.

  1. (RR)

    We say that f is real-rooted if all its roots are real.

  2. (LC)

    We say that f is log-concave if \(a_i^2 \ge a_{i-1}a_{i+1}\) for all i.

  3. (UN)

    We say that f is unimodal if \(a_0 \le a_1 \le \cdots \le a_k \ge \cdots \ge a_d\) for some k.

If all its coefficients are nonnegative, then these properties satisfy the implications

$$\begin{aligned} \mathrm{(RR)}\ \Rightarrow \ \mathrm{(LC)}\ \Rightarrow \ \mathrm{(UN)}. \end{aligned}$$

On the other hand, the polynomial f is \(\gamma \)-positive if f is palindromic and there are \(\gamma _0,\gamma _1,\ldots ,\gamma _{\lfloor d/2\rfloor } \ge 0\) such that \(f(x)=\sum _{i \ge 0}\gamma _ix^i (1+x)^{d-2i}\). The polynomial \(\sum _{i \ge 0}\gamma _ix^i\) is called the \(\gamma \)-polynomial of f. We can see that a \(\gamma \)-positive polynomial is real-rooted if and only if its \(\gamma \)-polynomial is real-rooted. If f is palindromic and real-rooted, then it is \(\gamma \)-positive. Moreover, if f is \(\gamma \)-positive, then it is unimodal. See, e.g., [2, 34] for details.

For a given lattice polytope, a fundamental problem within the field of Ehrhart theory is to determine if its \(h^*\)-polynomial is unimodal. One famous instance is given by reflexive polytopes that possess a regular unimodular triangulation.

Proposition 2.2

[5]Let \({\mathscr {P}}\subset {\mathbb R}^d\) be a reflexive polytope of dimension d. If P possesses a regular unimodular triangulation, then \(h^*({\mathscr {P}}, x)\) is unimodal.

It is known that if a reflexive polytope possesses a flag regular unimodular triangulation all of whose maximal simplices contain the origin, then the \(h^*\)-polynomial coincides with the h-polynomial of a flag triangulation of a sphere [5]. For the h-polynomial of a flag triangulation of a sphere, Gal [13] conjectured the following:

Conjecture 2.3

The h-polynomial of any flag triangulation of a sphere is \(\gamma \)-positive.

Classes of Anti-Blocking Polytopes and Unconditional Polytopes

In this section, we introduce several classes of anti-blocking polytopes and unconditional polytopes. Throughout this section, we associate each subset \(F \subset [d]\) with a (0, 1)-vector \({\mathbf{e}}_F \!=\! \sum _{i \in F} {\mathbf{e}}_i \!\in \! {\mathbb R}^d\), where each \({\mathbf{e}}_i\) is the ith unit coordinate vector in \({\mathbb R}^d\).

(0, 1)-Polytopes Arising from Simplicial Complexes

Let \(\Delta \) be a simplicial complex on the vertex set [d]. Then \(\Delta \) is a collection of subsets of [d] with \(\{i\} \in \Delta \) for all \(i \in [d]\) such that if \(F \in \Delta \) and \(F' \subset F\), then \(F' \in \Delta \). In particular \(\emptyset \in \Delta \) and \({\mathbf{e}}_{\emptyset }= \mathbf{0}\). Let \({\mathscr {P}}_\Delta \) denote the convex hull of \(\{ {\mathbf{e}}_F \in {\mathbb R}^d : F \in \Delta \}\). The following is an important observation.

Proposition 3.1

Let \({\mathscr {P}}\subset {\mathbb R}_{\ge 0}^d\) be a (0, 1)-polytope of dimension d. Then \({\mathscr {P}}\) is anti-blocking if and only if there exists a simplicial complex \(\Delta \) on [d] such that \({\mathscr {P}}= {\mathscr {P}}_\Delta \).

Stable Set Polytopes

Let G be a finite simple graph on the vertex set [d] and E(G) the set of edges of G. (A finite graph G is called simple if G possesses no loop and no multiple edge.) A subset \(W \subset [d]\) is called stable if, for all i and j belonging to W with \(i \ne j\), one has \(\{i,j\} \notin E(G)\). We remark that a stable set is often called an independent set. Let S(G) denote the set of all stable sets of G. One has \(\emptyset \in S(G)\) and \(\{ i \} \in S(G)\) for each \(i \in [d]\). The stable set polytope \({\mathscr {Q}}_G \subset {\mathbb R}^{d}\) of G is the (0, 1)-polytope defined by

$$\begin{aligned} {\mathscr {Q}}_G:={\text {conv}}{\{{\mathbf{e}}_W \in {\mathbb R}^d : W \in S(G) \}}. \end{aligned}$$

Then one has \(\dim {\mathscr {Q}}_G = d\). Since we can regard S(G) as a simplicial complex on [d], \({\mathscr {Q}}_G\) is an anti-blocking polytope.

Locally anti-blocking reflexive polytopes are characterized by stable set polytopes. A clique of G is a subset \(W \subset [d]\) that is a stable set of the complement graph \({\overline{G}}\) of G. The chromatic number of G is the smallest integer \(t \ge 1\) for which there exist stable sets \(W_{1}, \ldots , W_{t}\) of G with \([d] = W_{1} \cup \cdots \cup W_{t}\). A finite simple graph G is said to be perfect if, for any induced subgraph H of G including G itself, the chromatic number of H is equal to the maximal cardinality of cliques of H. See, e.g., [9] for details on graph theoretical terminology.

Proposition 3.2

[23]Let \({\mathscr {P}}\subset {\mathbb R}^d\) be a locally anti-blocking lattice polytope of dimension d. Then \({\mathscr {P}}\subset {\mathbb R}^d\) is reflexive if and only if, for each \(\varepsilon \in \{-1,1\}^d\), there exists a perfect graph \(G_{\varepsilon }\) on [d] such that \({\mathscr {P}}\cap {\mathbb R}^d_{\varepsilon }={\mathscr {Q}}_{G_{\varepsilon }}^{\pm } \cap {\mathbb R}^d_{\varepsilon }\).

Chain Polytopes and Enriched Chain Polytopes

Let \((P, <_P)\) be a partially ordered set (poset, for short) on [d]. A subset A of [d] is called an antichain of P if all i and j belonging to A with \(i \ne j\) are incomparable in P. In particular, the empty set \(\emptyset \) and each 1-element subset \(\{i\}\) are antichains of P. Let \({\mathscr {A}}(P)\) denote the set of antichains of P. In [37], Stanley introduced the chain polytope \({\mathscr {C}}_P\) of P defined by

$$\begin{aligned} {\mathscr {C}}_P:={\text {conv}}{\{ {\mathbf{e}}_A \in {\mathbb R}^d : A \in {\mathscr {A}}(P) \}}. \end{aligned}$$

It is known that chain polytopes are stable set polytopes. Indeed, let \(G_P\) be the finite simple graph on [d] such that \(\{i,j\} \in E(G_P)\) if and only if \(i <_P j\) or \(j <_P i\). We call \(G_P\) the comparability graph of P. It then follows that \({\mathscr {A}}(P)=S(G_P)\). Hence the chain polytope \({\mathscr {C}}_P\) is the stable set polytope \({\mathscr {Q}}_{G_P}\). Therefore, chain polytopes are anti-blocking polytopes. We remark that any comparability graph is perfect.

On the other hand, the enriched chain polytope \({\mathscr {C}}^{(e)}_P\) of P is the unconditional lattice polytope defined by \({\mathscr {C}}^{(e)}_P:={\mathscr {C}}_P^{\pm }\). In [30], it was shown that the Ehrhart polynomial of \({\mathscr {C}}^{(e)}_P\) coincides with a counting polynomial of left enriched P-partitions. We assume that P is naturally labeled. A map \(f:P \rightarrow {\mathbb Z}\setminus \{0\}\) is called an enriched P-partition [38] if, for all \(x, y \in P\) with \(x <_P y\), f satisfies

$$\begin{aligned} |f(x)| \le |f(y)|\quad \text {and}\quad |f(x)| = |f(y)|\ \Rightarrow \ f(y) > 0. \end{aligned}$$

A map \(f:P \rightarrow {\mathbb Z}\) is called a left enriched P-partition [33] if, for all \(x, y \in P\) with \(x <_P y\), f satisfies

$$\begin{aligned} |f(x)| \le |f(y)|\quad \text {and}\quad |f(x)| = |f(y)|\ \Rightarrow \ f(y) \ge 0. \end{aligned}$$

The symbol \(\Omega _P^{(\ell )}(m)\) will denote the number of left enriched P-partitions \(f:P \rightarrow {\mathbb Z}\) with \(|f(x)| \le m\) for any \(x \in P\), which is called the left enriched order polynomial of P.

Proposition 3.3

[30]Let P be a naturally labeled finite poset on [d]. Then one has

$$\begin{aligned} L_{{\mathscr {C}}^{(e)}_P}(m) =\Omega _P^{(\ell )} (m). \end{aligned}$$

Given a linear extension \(\pi = (\pi _1,\dots ,\pi _d)\) of a finite poset P on [d], a left peak of \(\pi \) is an index \(1 \le i \le d-1\) such that \(\pi _{i-1} <\pi _i > \pi _{i+1} \), where we set \(\pi _0 =0\). Let \(\mathrm{pk}^{(\ell )}(\pi )\) denote the number of left peaks of \(\pi \). Then the left peak polynomial \(W_{P}^{(\ell )} (x)\) of P is defined by

$$\begin{aligned} W_{P}^{(\ell )} (x) \,=\!\sum _{\pi \in {{\mathscr {L}}} (P)}\!\! x^{\ \mathrm{pk}^{(\ell )}(\pi )}, \end{aligned}$$

where \({\mathscr {L}}(P)\) is the set of linear extensions of P.

Proposition 3.4

[30]Let P be a naturally labeled finite poset on [d]. Then the \(h^*\)-polynomial of \({\mathscr {C}}^{(e)}_P\) is

$$\begin{aligned} h^*({\mathscr {C}}^{(e)}_P\!, x)= (x+1)^d W_{P}^{(\ell )} \biggl ( \frac{4x}{(x+1)^2} \biggl ). \end{aligned}$$

In particular, \(h^*({\mathscr {C}}^{(e)}_P, x)\) is \(\gamma \)-positive.

Note that if Q is a finite poset that is obtained from P by reordering the label, then \({\mathscr {C}}^{(e)}_P\) and \({\mathscr {C}}^{(e)}_Q\) are unimodularly equivalent. Hence the \(h^*\)-polynomials of enriched chain polytopes are always \(\gamma \)-positive.

Symmetric Edge Polytopes of Type B

Let G be a finite simple graph on [d]. We set

$$\begin{aligned} B_G :={\text {conv}}{\left( \{ \mathbf{0} , {\mathbf{e}}_1,\ldots ,{\mathbf{e}}_d\} \cup \{{\mathbf{e}}_i + {\mathbf{e}}_j : \{i,j\} \in E(G) \}\right) }. \end{aligned}$$

Then \(B_G = {\mathscr {P}}_\Delta \) where \(\Delta \) is a simplicial complex on [d] obtained by regarding G as a 1-dimensional simplicial complex. The symmetric edge polytope of type B of G is the unconditional lattice polytope defined by \({\mathscr {B}}_G:= B_G^{\pm }\).

Proposition 3.5

[31]Let G be a finite simple graph on [d]. Then \({\mathscr {B}}_G\) is reflexive if and only if G is bipartite.

A hypergraph is a pair \({\mathscr {H}}= (V, E)\), where \(E=\{e_1,\ldots ,e_n\}\) is a finite multiset of non-empty subsets of \(V=\{v_1,\ldots ,v_m\}\). Elements of V are called vertices and the elements of E are the hyperedges. Then we can associate \({\mathscr {H}}\) to a bipartite graph \({\text {Bip}}{\mathscr {H}}\) with a bipartition \(V \cup E\), such that \(\{v_i, e_j\}\) is an edge of \({\text {Bip}}{\mathscr {H}}\) if \(v_i \in e_j\). Assume that \({\text {Bip}}{\mathscr {H}}\) is connected. A hypertree in \({\mathscr {H}}\) is a function \(\mathbf{f}:E \rightarrow \{0,1,\ldots \}\) such that there exists a spanning tree \(\Gamma \) of \({\text {Bip}}{\mathscr {H}}\) whose vertices have degree \(\mathbf{f} (e) +1\) at each \(e \in E\). Then we say that \(\Gamma \) induces \(\mathbf{f}\). Let \(B_{\mathscr {H}}\) denote the set of all hypertrees in \({\mathscr {H}}\). A hyperedge \(e_j \in E\) is said to be internally active with respect to the hypertree \(\mathbf{f}\) if it is not possible to decrease \(\mathbf{f}(e_j)\) by 1 and increase \(\mathbf{f}(e_{j'})\), \(j' < j\), by 1 so that another hypertree results. We call a hyperedge internally inactive with respect to a hypertree if it is not internally active and denote the number of such hyperedges of \(\mathbf{f}\) by \({\overline{\iota }} (\mathbf{f}) \). Then the interior polynomial of \({\mathscr {H}}\) is the generating function \(I_{\mathscr {H}}(x)=\sum _{\mathbf{f} \in B_{\mathscr {H}}} x^{ {\overline{\iota }} (\mathbf{f})}\). It is known [22, Prop. 6.1] that \({\text {deg}}I_{\mathscr {H}}(x)\le \min {\{|V|,|E|\}}-1\). If \(G={\text {Bip}}{\mathscr {H}}\), then we set \(I_G(x)=I_{\mathscr {H}}(x)\).

Assume that G is a bipartite graph with a bipartition \(V_1 \cup V_2 =[d]\). Then let \({\widetilde{G}}\) be a connected bipartite graph on \([d+2]\) whose edge set is

$$\begin{aligned} E({\widetilde{G}}) = E(G) \cup \{ \{i, d+1\} : i \in V_1\} \cup \{ \{j, d+2\} : j \in V_2 \cup \{d+1\}\}. \end{aligned}$$

Proposition 3.6

[31]Let G be a bipartite graph on [d]. Then the \(h^*\)-polynomial of the reflexive polytope \({\mathscr {B}}_G\) is

$$\begin{aligned} h^*({\mathscr {B}}_G, x)= (x+1)^d I_{{\widetilde{G}}} \biggl (\frac{4x}{(x+1)^2} \biggr ). \end{aligned}$$

In particular, \(h^*({\mathscr {B}}_G, x)\) is \(\gamma \)-positive.

\(h^*\)-Polynomials of Locally Anti-Blocking Lattice Polytopes

In the present section, we prove Theorem 1.1, that is, a formula for the \(h^*\)-polynomials of locally anti-blocking lattice polytopes in terms of that of unconditional lattice polytopes. Given a subset \(J=\{j_1,\dots , j_r\}\) of [d], let

$$\begin{aligned} \pi _J :{\mathbb R}^d \rightarrow {\mathbb R}^r, \qquad \pi _J((x_1,\dots ,x_d)) = (x_{j_1},\dots ,x_{j_r}) \end{aligned}$$

denote the projection map. (Here \(\pi _\emptyset \) is the zero map.)

Proposition 4.1

Let \({\mathscr {P}}\subset {\mathbb R}_{\ge 0}^d\) be an anti-blocking lattice polytope. Then we have

$$\begin{aligned} h^*({\mathscr {P}}^\pm , x)\,=\,\sum _{j=0}^d 2^j (x-1)^{d-j}\!\!\sum _{J \subset [d], \, |J| = j}\!\! h^*(\pi _J ({\mathscr {P}}), x). \end{aligned}$$

Proof

The proof is similar to the discussion in [31, proof of Prop. 3.1]. The intersection of \({\mathscr {P}}^\pm \cap {\mathbb R}_\varepsilon ^d\) and \({\mathscr {P}}^\pm \cap {\mathbb R}_{\varepsilon '}^d\) is of dimension \(d-1\) if and only if \(\varepsilon - \varepsilon ' \in \{\pm 2 {\mathbf{e}}_1, \ldots , \pm 2 {\mathbf{e}}_d\}\). Moreover, if \(\varepsilon - \varepsilon ' = 2 {\mathbf{e}}_k\), then

$$\begin{aligned} ({\mathscr {P}}^\pm \cap {\mathbb R}_\varepsilon ^d) \cap ({\mathscr {P}}^\pm \cap {\mathbb R}_{\varepsilon '}^d)={\mathscr {P}}^\pm \cap {\mathbb R}_\varepsilon ^d \cap {\mathbb R}_{\varepsilon '}^d\simeq & {} \,\,\pi _{[d] \setminus \{k\}} ({\mathscr {P}}^\pm ) \cap {\mathbb R}_{ \pi _{[d] \setminus \{k\}} (\varepsilon )}^{d-1}\\\simeq & {} \,\pi _{[d] \setminus \{k\}} ({\mathscr {P}}). \end{aligned}$$

Hence the Ehrhart polynomial \(L_{{\mathscr {P}}^\pm }(m)\) satisfies the following:

$$\begin{aligned} L_{{\mathscr {P}}^\pm }(m)\, =\, \sum _{j=0}^d2^j (-1)^{d-j}\!\!\sum _{J \subset [d], \, |J| = j}\!\! L_{\pi _J ({\mathscr {P}})}(m). \end{aligned}$$

Thus the Ehrhart series satisfies

$$\begin{aligned} \frac{h^*({\mathscr {P}}^\pm , x)}{(1-x)^{d+1}}\,=\,\sum _{j=0}^d2^j (-1)^{d-j}\!\!\sum _{J \subset [d], \, |J| = j}\!\frac{h^*(\pi _J ({\mathscr {P}}), x)}{(1-x)^{j+1}}, \end{aligned}$$

as desired. \(\square \)

We now prove Theorem 1.1.

Proof of Theorem 1.1

Given \(J = \{j_1,\dots , j_r\} \subset [d]\) and \(\varepsilon \in \{-1,1\}^{r}\), let

$$\begin{aligned} {\mathbb R}_{J, \varepsilon }^d \,=\,\{ {\mathbf{x}}= (x_1,\ldots , x_d) \in {\mathbb R}^d :\pi _J({\mathbf{x}}) \in {\mathbb R}_\varepsilon ^r\ \text {and}\ x_j =0\ \text {for all}\ j \notin J \}. \end{aligned}$$

It then follows that \({\mathscr {P}}\cap {\mathbb R}_{J, \varepsilon }^d\) is equal to \(\pi _J({\mathscr {P}}_{\varepsilon '})^\pm \cap {\mathbb R}_\varepsilon ^r\), where \(\pi _J(\varepsilon ') = \varepsilon \). Note that, given \(J = \{j_1,\dots , j_r\} \subset [d]\) and \(\varepsilon \in \{-1,1\}^{r}\), we have \(| \{ \varepsilon ' \in \{-1,1\}^d : \pi _J(\varepsilon ') = \varepsilon \} |= 2^{d-r}\). Thus

$$\begin{aligned} h^*({\mathscr {P}}, x)&\,=\,\sum _{j=0}^d (x-1)^{d-j}\!\!\sum _{J \subset [d],\, |J| = j }\ \sum _{\varepsilon \in \{-1,1\}^j}\!\!h^*({\mathscr {P}}\cap {\mathbb R}_{J, \varepsilon }^d, x)\\&\,=\,\sum _{j=0}^d (x-1)^{d-j}\!\!\sum _{\varepsilon \in \{-1,1\}^d}\ \sum _{J \subset [d],\, |J| = j }\!\!\frac{h^*(\pi _J({\mathscr {P}}_{\varepsilon }), x)}{2^{d-j}}\\&\,=\,\frac{1}{2^{d}}\sum _{\varepsilon \in \{-1,1\}^d}\ \sum _{j=0}^d 2^j (x-1)^{d-j}\!\!\sum _{J \subset [d],\, |J| = j }\!\!h^*(\pi _J({\mathscr {P}}_{\varepsilon }), x)\\&=\frac{1}{2^{d}}\sum _{\varepsilon \in \{-1,1\}^d}\!\!h^*({\mathscr {P}}_\varepsilon ^\pm , x) \end{aligned}$$

by Proposition 4.1. \(\square \)

Combining Theorem 1.1 with Propositions 3.4 and 3.6, we have

Corollary 4.2

Let \({\mathscr {P}}\subset {\mathbb R}^d\) be a locally anti-blocking reflexive polytope. If every \({\mathscr {P}}\cap {\mathbb R}_\varepsilon ^d\) is the intersection of \({\mathbb R}_\varepsilon ^d\) and either an enriched chain polytope or a symmetric edge reflexive polytope of type B, then the \(h^*\)-polynomial of \({\mathscr {P}}\) is \(\gamma \)-positive.

Finally, we conjecture the following.

Conjecture 4.3

The \(h^*\)-polynomial of any locally anti-blocking reflexive polytope is \(\gamma \)-positive.

Thanks to Theorem 1.1 and Proposition 3.2, in order to prove Conjecture 4.3, it is enough to study unconditional lattice polytopes \({\mathscr {Q}}_G^\pm \) where \({\mathscr {Q}}_G\) is the stable set polytope of a perfect graph G.

Symmetric Edge Polytopes of Type A

Let G be a finite simple graph on the vertex set [d] and the edge set E(G). The symmetric edge polytope \({{\mathscr {A}}}_G \subset {\mathbb R}^d\) of type A is the convex hull of the set

$$\begin{aligned} A(G) = \{ \pm ( {\mathbf{e}}_i - {\mathbf{e}}_j ) \in {\mathbb R}^d: \{ i, j\} \in E(G) \}. \end{aligned}$$

The polytope \({{\mathscr {A}}}_G\) is introduced in [26, 28] and called a “symmetric edge polytope of G.”

Example 5.1

Let G be a tree on [d]. Then \({\mathscr {A}}_G\) is unimodularly equivalent to a \((d-1)\)-dimensional cross polytope. Hence we have \(h^*({\mathscr {A}}_G,x)= (x+1)^{d-1}\).

It is known [26, Prop. 4.1] that the dimension of \({\mathscr {A}}_G\) is \(d-1\) if and only if G is connected. Higashitani [20] proved that \({{\mathscr {A}}}_G\) is simple if and only if \({{\mathscr {A}}}_G\) is smooth Fano if and only if G contains no even cycles. It is known [26, 28] that \({{\mathscr {A}}}_G\) is unimodularly equivalent to a reflexive polytope having a regular unimodular triangulation. In particular, the \(h^*\)-polynomial of \({{\mathscr {A}}}_{G}\) is palindromic and unimodal. For a complete bipartite graph \(K_{\ell , m}\), it is known [21] that the \(h^*\)-polynomial of \({{\mathscr {A}}}_{K_{\ell , m}}\) is real-rooted and hence \(\gamma \)-positive.

Recursive Formulas for \(h^*\)-Polynomials

In this section, we give several recursive formulas of \(h^*\)-polynomials of \({\mathscr {A}}_G\) when G belongs to certain classes of graphs. By the following fact, we may assume that G is 2-connected if needed.

Proposition 5.2

Let G be a graph and let \(G_1,\ldots , G_s\) be 2-connected components of G. Then the \(h^*\)-polynomial of \({{\mathscr {A}}}_{G}\) satisfies

$$\begin{aligned} h^*({\mathscr {A}}_G,x)=h^*({\mathscr {A}}_{G_1},x) \cdots h^*({\mathscr {A}}_{G_s},x). \end{aligned}$$

Proof

Since \({\mathscr {A}}_G\) is the free sum of reflexive polytopes \({\mathscr {A}}_{G_1}, \ldots , {\mathscr {A}}_{G_s}\), a desired conclusion follows from [4, Thm. 1]. \(\square \)

The suspension \({\widehat{G}}\) of a graph G is the graph on the vertex set \([d+1]\) and the edge set

$$\begin{aligned} E(G)\cup \{\{i, d+1\} : i \in [d]\}. \end{aligned}$$

We now study the \(h^*\)-polynomial of \({{\mathscr {A}}}_{{\widehat{G}}}\). Given a subset \(S \subset [d]\),

$$\begin{aligned} E_S := \{ e \in E(G) : |e \cap S| =1 \} \end{aligned}$$

is called a cut of G. For example, we have \(E_\emptyset = E_{[d]} = \emptyset \). In general, it follows that \(E_S = E_{[d] \setminus S}\). We identify \(E_S\) with the subgraph of G on the vertex set [d] and the edge set \(E_S\). By definition, \(E_S\) is a bipartite graph. Let \({\text {Cut}}(G)\) be the set of all cuts of G. Note that \(|{{\text {Cut}}(G)}| = 2^{d-1}\). From Theorem 1.1 and Proposition 3.6, we have the following.

Theorem 5.3

Let G be a finite graph on [d]. Then \({{\mathscr {A}}}_{{\widehat{G}}}\) is unimodularly equivalent to a locally anti-blocking reflexive polytope whose \(h^*\)-polynomial is

$$\begin{aligned} h^*({{\mathscr {A}}}_{{\widehat{G}}}, x)=\frac{1}{2^{d-1}}\sum _{H \in {\text {Cut}}(G)}\!\!h^*({{\mathscr {B}}}_H, x)=(x+1)^d f_G \bigg ( \frac{4x}{(x+1)^2} \bigg ), \end{aligned}$$

where

$$\begin{aligned} f_G (x)= \frac{1}{2^{d-1}} \sum _{H \in {\text {Cut}}(G)} \!\!I_{{\widetilde{H}}} (x). \end{aligned}$$

In particular, \(h^*({{\mathscr {A}}}_{{\widehat{G}}}, x)\) is \(\gamma \)-positive. Moreover, \(h^*({{\mathscr {A}}}_{{\widehat{G}}}, x)\) is real-rooted if and only if \(f_G(x)\) is real-rooted.

Proof

Let \({{\mathscr {P}}} \subset {\mathbb R}^d\) be the convex hull of

$$\begin{aligned} \{ \pm {\mathbf{e}}_1, \dots , \pm {\mathbf{e}}_d\}\cup \{ \pm ( {\mathbf{e}}_i - {\mathbf{e}}_j ): \{ i, j\} \in E(G) \}. \end{aligned}$$

Then \({{\mathscr {A}}}_{{\widehat{G}}}\) is lattice isomorphic to \({\mathscr {P}}\). Given \(\varepsilon = (\varepsilon _1,\ldots , \varepsilon _d) \in \{-1,1\}^d\), let \(S_\varepsilon = \{ i \in [d] : \varepsilon _i = 1\}\). Then \({{\mathscr {P}}} \cap {\mathbb R}_\varepsilon ^d\) is the convex hull of

$$\begin{aligned} \{\mathbf{0}\} \cup \{ \varepsilon _i {\mathbf{e}}_i : i \in [d] \}\cup \{ {\mathbf{e}}_i - {\mathbf{e}}_j : \{ i, j\} \in E_{S_\varepsilon } ,\, i \in S_\varepsilon \}. \end{aligned}$$

Hence \({{\mathscr {P}}} \cap {\mathbb R}_\varepsilon ^d = {\mathscr {B}}_{E_{S_\varepsilon }} \cap {\mathbb R}_\varepsilon ^d\). Thus \({\mathscr {P}}\) is a locally anti-blocking polytope and

$$\begin{aligned} h^*({{\mathscr {A}}}_{{\widehat{G}}}, x)=\frac{1}{2^{d-1}}\sum _{H \in {\text {Cut}}(G)}\!\!h^*({{\mathscr {B}}}_H, x) \end{aligned}$$

by Theorem 1.1. \(\square \)

Let G be a graph and let \(e=\{i,j\}\) be an edge of G. Then the graph G/e obtained by the procedure

  1. (i)

    Delete e and identify the vertices i and j

  2. (ii)

    Delete the multiple edges that may be created while (i)

is called the graph obtained from G by contracting the edge e. Next, we will show that, for any bipartite graph G and \(e \in E(G)\), \(h^*({{\mathscr {A}}}_G, x)\) is \(\gamma \)-positive if and only if so is \(h^*({\mathscr {A}}_{G/e}, x)\). In order to show this fact, we need the theory of Gröbner bases of toric ideals. Given a graph G on the vertex set [d] and the edge set \(E(G)=\{e_1 ,\dots , e_n\}\), let

$$\begin{aligned} {\mathscr {R}}=K[t_1, t_1^{-1}\!, \dots , t_d, t_d^{-1}\!,s ] \end{aligned}$$

be the Laurent polynomial ring over a field K and let

$$\begin{aligned} {\mathscr {S}}=K[x_1, \dots , x_n, y_1, \dots , y_n, z] \end{aligned}$$

be the polynomial ring over K. We define the ring homomorphism \(\pi :{\mathscr {S}}\rightarrow {\mathscr {R}}\) by setting \(\pi (z) = s\), \(\pi (x_k) = t_i t_j^{-1} s\) and \(\pi (y_k) = t_i^{-1} t_j s\) if \(e_k = \{i,j\} \in E(G)\) and \(i<j\). The toric ideal \(I_{{\mathscr {A}}_G}\) of \({\mathscr {A}}_G\) is the kernel of \(\pi \). (See, e.g., [14] for details on toric ideals and Gröbner bases.) We now recall the notation given in [21]. For any oriented edge \(e_i\), let \(p_i\) denote the corresponding variable, i.e., \(p_i = x_i\) or \(p_i = y_i\) depending on the orientation, and let \(\{p_i, q_i\} = \{x_i, y_i\}\). Let \({\mathscr {G}}(G)\) be the set of all binomials f satisfying one of the following:

$$\begin{aligned} f\,=\, \prod _{e_i \in I} p_i \,-\! \prod _{e_i \in C \setminus I} q_i, \end{aligned}$$
(1)

where C is an even cycle in G of length 2k with a fixed orientation, and I is a k-subset of C such that \(e_\ell \notin I\) for \(\ell =\min {\{i:e_i\in C\}}\);

$$\begin{aligned} f\,=\, \prod _{e_i \in I} p_i - z\prod _{e_i \in C \setminus I} q_i, \end{aligned}$$
(2)

where C is an odd cycle in G of length \(2k+1\) and I is a \((k+1)\)-subset of C;

$$\begin{aligned} f = x_i y_i - z^2, \end{aligned}$$
(3)

where \(1 \le i \le n\). Then \({\mathscr {G}}(G)\) is a Gröbner basis of \(I_{{\mathscr {A}}_G}\) with respect to a reverse lexicographic order < induced by the ordering \(z< x_1< y_1< \cdots< x_n < y_n\) [21, Prop. 3.8]. Here the initial monomial of each binomial is the first monomial. Using this Gröbner basis, we have the following.

Proposition 5.4

Let G be a bipartite graph on [d] and let \(e \in E(G)\). Then we have

$$\begin{aligned} h^*({\mathscr {A}}_G, x) = (x+1) h^*({\mathscr {A}}_{G/e}, x). \end{aligned}$$

Proof

Let \(E(G)=\{e_1 ,\dots , e_n\}\) with \(e = e_1 = \{i,j\}\). Since G is a bipartite graph, the Gröbner basis \({\mathscr {G}}(G)\) above consists of the binomials of the form (1) and (3).

Since G has no triangles, the procedure (ii) does not occur when we contract e of G. Hence \(E(G/e) = \{e_2' , \dots , e_n' \}\) where \(e_k'\) is obtained from \(e_k\) by identifying i with j. Let \(G'\) be a graph obtained by adding an edge \(e_1' = \{d+1, d+2\}\) to the graph G/e. Then \({\mathscr {G}}(G')\) consists of all binomials f satisfying one of the following:

$$\begin{aligned} f\,=\, \prod _{e_i \in I} p_i - \prod _{e_i \in C \setminus I} q_i, \end{aligned}$$

where C is an even cycle in G of length 2k with a fixed orientation and \(e_1 \notin C\), and I is a k-subset of C such that \(e_\ell \notin I\) for \(\ell = \min {\{i : e_i \in C\}}\);

$$\begin{aligned} f\,=\, \prod _{e_i \in I} p_i \,-\, z\prod _{e_i \in C \setminus I} q_i, \end{aligned}$$

where \(C \cup \{e_1\}\) is an even cycle in G of length \(2k+2\) and I is a \((k+1)\)-subset of C;

$$\begin{aligned} f = x_i y_i - z^2, \end{aligned}$$

where \(1 \le i \le n\). Hence \(\{ {\text {in}}_< (f) : f \in {\mathscr {G}}(G) \} = \{{\text {in}}_< (f) : f \in {\mathscr {G}}(G') \}\). By a similar argument as in the proof of [19, Thm. 3.1], it follows that

$$\begin{aligned} h^*({\mathscr {A}}_G, x) = h^*( {\mathscr {A}}_{G'}, x) = h^*({\mathscr {A}}_{\{e_1'\}}, x) h^*({\mathscr {A}}_{G/e}, x)= (x+1) h^*({\mathscr {A}}_{G/e}, x) , \end{aligned}$$

as desired. \(\square \)

From Theorem 5.3, Propositions 5.2 and 5.4 we have the following immediately.

Corollary 5.5

Let G be a bipartite graph on [d]. Then we have that:

  1. (a)

    The \(h^*\)-polynomial \(h^*({\mathscr {A}}_{{\widetilde{G}}}, x) = (x+1) h^*({\mathscr {A}}_{{\widehat{G}}}, x)\) is \(\gamma \)-positive.

  2. (b)

    If G is obtained by gluing bipartite graphs \(G_1\) and \(G_2\) along with an edge e, then

    $$\begin{aligned} h^*({\mathscr {A}}_G, x)&=(x+1) h^*({\mathscr {A}}_{G/e}, x) \\&=(x+1) h^*({\mathscr {A}}_{G_1/e}, x) h^*({\mathscr {A}}_{G_2/e}, x)\\&=h^*({\mathscr {A}}_{G_1}, x) h^*({\mathscr {A}}_{G_2}, x)/(x+1). \end{aligned}$$

Remark

Corollary 5.5 (b) was recently generalized in [8, Thm. 4.17].

Pseudo-Symmetric Simplicial Reflexive Polytopes

A lattice polytope \({\mathscr {P}}\subset {\mathbb R}^d\) is called pseudo-symmetric if there exists a facet \({\mathscr {F}}\) of \({\mathscr {P}}\) such that \(-{\mathscr {F}}\) is also a facet of \({\mathscr {P}}\). Nill [27] proved that any pseudo-symmetric simplicial reflexive polytope \({\mathscr {P}}\) is a free sum of \({\mathscr {P}}_1, \dots , {\mathscr {P}}_s\), where each \({\mathscr {P}}_i\) is one of the following:

  • cross polytope;

  • del Pezzo polytope \(V_{2m} = {\text {conv}}{( \pm {\mathbf{e}}_1,\dots ,\pm {\mathbf{e}}_{2m}, \pm ( {\mathbf{e}}_1 + \dots + {\mathbf{e}}_{2m}) )}\);

  • pseudo-del Pezzo polytope \({\widetilde{V}}_{2m} = {\text {conv}}{( \pm {\mathbf{e}}_1,\dots ,\pm {\mathbf{e}}_{2m},-{\mathbf{e}}_1 - \dots - {\mathbf{e}}_{2m} )}\).

Note that a del Pezzo polytope is unimodularly equivalent to \({\mathscr {A}}_{C_{2m+1}}\) where \(C_{2m+1}\) is an odd cycle of length \(2m+1\) (see [20]). The \(h^*\)-polynomial of \({\mathscr {A}}_{C_d}\) was essentially studied in the following papers (see also the OEIS sequence A204621):

  • Conway and Sloane [6, p. 2379] computed \(h^*({\mathscr {A}}_{C_d},x)\) for small d by using results of O’Keeffe [32] and gave a conjecture on the \(\gamma \)-polynomial of \(h^*({\mathscr {A}}_{C_d},x)\) (coincides with the \(\gamma \)-polynomial in Proposition 5.7 below).

  • General formulas for the coefficients of \(h^*({\mathscr {A}}_{C_d},x)\) were given in Ohsugi–Shibata [29] and Wang–Yu [40].

In order to give the \(h^*\)-polynomial of \({\widetilde{V}}_{2m}\), we need the following lemma.

Lemma 5.6

Let G be a connected graph. Suppose that an edge \(e=\{i,j\} \) of G is not a bridge. Let \({\mathscr {P}}_e\) be the convex hull of \(A(G) \setminus \{ {\mathbf{e}}_i - {\mathbf{e}}_j \}\). Then we have

$$\begin{aligned} h^*({\mathscr {P}}_e,x) = \frac{h^*({\mathscr {A}}_G,x) + h^*({\mathscr {A}}_{G \setminus e},x)}{2}, \end{aligned}$$

where \(G \setminus e\) is the graph obtained by deleting e from G.

Proof

Note that \({\mathscr {A}}_{G \setminus e} \subset {\mathscr {P}}_e \subset {\mathscr {A}}_G\). Since G is connected and e is not a bridge of G, the dimension of both \({\mathscr {A}}_G\) and \({\mathscr {A}}_{G \setminus e}\) is \(d-1\). Let \({\mathscr {P}}_e'\) denote the convex hull of \(A(G) \setminus \{ - {\mathbf{e}}_i + {\mathbf{e}}_j \}\), which is unimodularly equivalent to \({\mathscr {P}}_e\). Then \({\mathscr {A}}_G\) and \({\mathscr {P}}_e\) are decomposed into the following disjoint union:

$$\begin{aligned} {\mathscr {A}}_G&={\mathscr {A}}_{G \setminus e} \cup ({\mathscr {P}}_e \setminus {\mathscr {A}}_{G \setminus e}) \cup ({\mathscr {P}}_e' \setminus {\mathscr {A}}_{G \setminus e}),\\ {\mathscr {P}}_e&={\mathscr {A}}_{G \setminus e} \cup ({\mathscr {P}}_e \setminus {\mathscr {A}}_{G \setminus e}). \end{aligned}$$

Since \({\mathscr {P}}_e \setminus {\mathscr {A}}_{G \setminus e}\) is unimodularly equivalent to \({\mathscr {P}}_e' \setminus {\mathscr {A}}_{G \setminus e}\), we have a desired conclusion. \(\square \)

The \(h^*\)-polynomials of \(V_{2m}\) and \({\widetilde{V}}_{2m}\) are as follows:

Proposition 5.7

Let \(C_d\) denote a cycle of length \(d \ge 3\) and let \(1 \le m \in {\mathbb Z}\). Then we have

$$\begin{aligned} h^*({\mathscr {A}}_{C_d},x)&= \sum _{i=0}^{\lfloor ({d-1})/{2} \rfloor }\left( {\begin{array}{c}2i\\ i\end{array}}\right) x^i (x+1)^{d-2i-1},\\ h^*(V_{2m},x)&=\sum _{i=0}^m\left( {\begin{array}{c}2i\\ i\end{array}}\right) x^i (x+1)^{2m-2i},\\ h^*({\widetilde{V}}_{2m},x)&= (x+1)^{2m}+ \sum _{i=1}^m\left( {\begin{array}{c}2i-1\\ i-1\end{array}}\right) x^i (x+1)^{2m-2i}. \end{aligned}$$

In particular, the \(h^*\)-polynomials of \({\mathscr {A}}_{C_d}\), \(V_{2m}\), and \({\widetilde{V}}_{2m}\) are \(\gamma \)-positive.

Proof

The proof for \(C_d\) is by induction on d. First, we have \(h^*({\mathscr {A}}_{C_{3}},x) = x^2 + 4x +1 = (x+1)^2 + \left( {\begin{array}{c}2\\ 1\end{array}}\right) x\). If \(d \ge 4\) is even, then

$$\begin{aligned} h^*({\mathscr {A}}_{C_{d}},x)&= (x + 1) h^*({\mathscr {A}}_{C_{d-1}},x)\\&=\sum _{i=0}^{({d-2})/{2}}\left( {\begin{array}{c}2i\\ i\end{array}}\right) x^i (x+1)^{d-2i-1}\,=\!\sum _{i=0}^{\lfloor ({d-1})/{2} \rfloor }\left( {\begin{array}{c}2i\\ i\end{array}}\right) x^i (x+1)^{d-2i-1}. \end{aligned}$$

Moreover, if \(d = 2m +1\), \(2 \le m \in {\mathbb Z}\), then the coefficient of \(x^m\) in

$$\begin{aligned} \sum _{i=0}^{({d-1})/{2}}\left( {\begin{array}{c}2i\\ i\end{array}}\right) x^i (x+1)^{d-2i-1}= (x+1) h^*({\mathscr {A}}_{C_{d-1}},x)+ \left( {\begin{array}{c}2m\\ m\end{array}}\right) x^m \end{aligned}$$

is

$$\begin{aligned} \sum _{i=0}^m \left( {\begin{array}{c}2i\\ i\end{array}}\right) \left( {\begin{array}{c}2m-2i\\ m-i\end{array}}\right) = 4^m = 2^{d-1}, \end{aligned}$$

and the other coefficient is arising from \((x+1) h^*({\mathscr {A}}_{C_{d-1}},x)\). By a recursive formula in [29, Thm. 2.3], we have

$$\begin{aligned} h^*({\mathscr {A}}_{C_d},x)\,=\! \sum _{i=0}^{({d-1})/{2}}\left( {\begin{array}{c}2i\\ i\end{array}}\right) x^i (x+1)^{d-2i-1}. \end{aligned}$$

Since \(V_{2m}\) is unimodularly equivalent to \({\mathscr {A}}_{C_{2m+1}}\), we have \(h^*(V_{2m},x) = h^*({\mathscr {A}}_{C_{2m+1}},x)\). By Lemma 5.6, it follows that

$$\begin{aligned} h^*({\widetilde{V}}_{2m},x)&=\frac{h^*({\mathscr {A}}_{C_{2m+1}},x) + h^*({\mathscr {A}}_{P_{2m+1}},x)}{2}\\&=\frac{1}{2} \sum _{i=0}^m\left( {\begin{array}{c}2i\\ i\end{array}}\right) x^i (x+1)^{2m-2i} + \frac{(x+1)^{2m}}{2}\\&=(x+1)^{2m}+ \sum _{i=1}^m\left( {\begin{array}{c}2i-1\\ i-1\end{array}}\right) x^i (x+1)^{2m-2i}. \end{aligned}$$

\(\square \)

Thus it turns out that any pseudo-symmetric simplicial reflexive polytope is a free sum of reflexive polytopes whose \(h^*\)-polynomials are \(\gamma \)-positive. By [4, Thm. 1], we have the following.

Theorem 5.8

The \(h^*\)-polynomial of any pseudo-symmetric simplicial reflexive polytope is \(\gamma \)-positive.

Proof

From results by Nill [27], any pseudo-symmetric simplicial reflexive polytope is a free sum of cross polytopes, del Pezzo polytopes, and pseudo-del Pezzo polytopes. On the other hand, by [4, Thm. 1], the \(h^*\)-polynomial of a free sum of reflexive polytopes \({\mathscr {P}}_1,\ldots ,{\mathscr {P}}_s\) is equal to the product of \(h^*\)-polynomials of \({\mathscr {P}}_1,\ldots , {\mathscr {P}}_s\). Hence, by Example 5.1 and Proposition 5.7, it follows that the \(h^*\)-polynomial of any pseudo-symmetric simplicial reflexive polytope is \(\gamma \)-positive. \(\square \)

Classes of Graphs with \(h^*({{\mathscr {A}}}_G, x)\) Being \(\gamma \)-Positive

With the results of the present section one can show that, for example, \(h^*({{\mathscr {A}}}_G, x)\) is \(\gamma \)-positive if one of the following holds:

  • \(G = {\widehat{H}}\) for some graph H (e.g., G is a complete graph, a wheel graph);

  • \(G = {\widetilde{H}}\) for some bipartite graph H (e.g., G is a complete bipartite graph);

  • G is a cycle;

  • G is an outerplanar bipartite graph.

Moreover, one can compute \(h^*({{\mathscr {A}}}_G, x)\) explicitly in some cases. We give such calculations for some known formulas (for complete [1] and complete bipartite graphs [21]).

Example 5.9

[1] By Theorem 5.3, we have

$$\begin{aligned} h^*({{\mathscr {A}}}_{K_d}, x) \,=\, h^*({{\mathscr {A}}}_{{\widehat{K}}_{d-1}}, x)\,=\,\frac{(x+1)^{d-1} }{2^{d-2}}\!\! \sum _{H \in {\text {Cut}}(K_{d-1})}\!\! I_{{\widetilde{H}}} \biggl (\frac{4x}{(x+1)^2}\biggr ). \end{aligned}$$

If the edge set of \(H \in {\text {Cut}}(K_{d-1})\) is \(E_S\) with \(S \subset [d-1]\), then H is a complete bipartite graph \(K_{|S|, d-1-|S|}\) and

$$\begin{aligned} I_{{\widetilde{H}}}(x) = \sum _{i\ge 0} \left( {\begin{array}{c}|S|\\ i\end{array}}\right) \left( {\begin{array}{c}d-|S|-1\\ i\end{array}}\right) x^i. \end{aligned}$$

(Here \(K_{0,d-1}\) denotes an empty graph.) It then follows that

$$\begin{aligned} h^*({{\mathscr {A}}}_{K_d}, x)&=\frac{1}{2^{d-1}}\sum _{k=0}^{d-1}\left( {\begin{array}{c}d-1\\ k\end{array}}\right) \sum _{i = 0}^{\lfloor ({d-1})/{2} \rfloor } 4^i \left( {\begin{array}{c}k\\ i\end{array}}\right) \left( {\begin{array}{c}d-k-1\\ i\end{array}}\right) x^i (x+1)^{d-1-2i}\\&=\frac{1}{2^{d-1}}\sum _{i = 0}^{\lfloor ({d-1})/{2} \rfloor }\!4^i x^i (x+1)^{d-1-2i}\sum _{k=i}^{d-i-1}\left( {\begin{array}{c}d-1\\ k\end{array}}\right) \left( {\begin{array}{c}k\\ i\end{array}}\right) \left( {\begin{array}{c}d-k-1\\ i\end{array}}\right) \\&=\frac{1}{2^{d-1}}\sum _{i = 0}^{\lfloor ({d-1})/{2} \rfloor }\!4^i x^i (x+1)^{d-1-2i}\sum _{k=i}^{d-i-1}\left( {\begin{array}{c}d-1\\ 2i\end{array}}\right) \left( {\begin{array}{c}2i\\ i\end{array}}\right) \left( {\begin{array}{c}d-1-2i\\ k-i\end{array}}\right) \\&=\frac{1}{2^{d-1}}\sum _{i = 0}^{\lfloor ({d-1})/{2} \rfloor }\!4^i x^i (x+1)^{d-1-2i}2^{d-1-2i}\left( {\begin{array}{c}d-1\\ 2i\end{array}}\right) \left( {\begin{array}{c}2i\\ i\end{array}}\right) \\&=\sum _{i = 0}^{\lfloor ({d-1})/{2} \rfloor }\left( {\begin{array}{c}d-1\\ 2i\end{array}}\right) \left( {\begin{array}{c}2i\\ i\end{array}}\right) x^i (x+1)^{d-1-2i} . \end{aligned}$$

Example 5.10

[21]   Let \(G=K_{m,n}\). Then \({\widetilde{G}} = K_{m+1,n+1}\) and

$$\begin{aligned} h^*({{\mathscr {A}}}_{K_{m+1,n+1}}, x)&=\,(x+1) h^*({{\mathscr {A}}}_{{\widehat{K}}_{m,n}}, x)\\&= \frac{(x+1)^{m+n+1} }{2^{m+n-1}} \!\!\sum _{H \in {\text {Cut}}(K_{m,n})} \!\! I_{{\widetilde{H}}} \biggl (\frac{4x}{(x+1)^2}\biggr ). \end{aligned}$$

Let \(V_1 \cup V_2\) be the partition of the vertex set of \(K_{m,n}\), where \(|V_1|=m\) and \(|V_2|=n\). If the edge set of \(H \in {\text {Cut}}(K_{m,n})\) is \(E_S\) with \(S \subset [m+n]\), then H is the disjoint union of two complete bipartite graphs \(K_{k, \ell }\) and \(K_{m-k,n-\ell }\), and hence

$$\begin{aligned} I_{{\widetilde{H}}}(x) = \sum _{i\ge 0} \left( {\begin{array}{c}k\\ i\end{array}}\right) \left( {\begin{array}{c}\ell \\ i\end{array}}\right) x^i \times \sum _{j\ge 0} \left( {\begin{array}{c}m-k\\ j\end{array}}\right) \left( {\begin{array}{c}n-\ell \\ j\end{array}}\right) x^j , \end{aligned}$$

where \(k=|V_1 \cap S|\) and \(\ell = n- |V_2 \cap S|\). It then follows that

$$\begin{aligned} h^*({{\mathscr {A}}}_{K_{m+1,n+1}}, x)&=\frac{x+1}{2^{m+n}}\sum _{k=0}^m\sum _{\ell =0}^n\left( {\begin{array}{c}m\\ k\end{array}}\right) \left( {\begin{array}{c}n\\ \ell \end{array}}\right) \sum _{i=0}^{\min {(k,\ell )}} 4^i \left( {\begin{array}{c}k\\ i\end{array}}\right) \left( {\begin{array}{c}\ell \\ i\end{array}}\right) x^i (x+1)^{k+\ell -2i} \\&\quad \times \sum _{j=0}^{\min {(m-k,n-\ell )}} 4^j \left( {\begin{array}{c}m-k\\ j\end{array}}\right) \left( {\begin{array}{c}n-\ell \\ j\end{array}}\right) x^j (x+1)^{m+n-k-\ell -2j}\\&=\frac{1}{2^{m+n}}\sum _{i,j\ge 0}4^{i+j}x^{i+j} (x+1)^{n+m-2(i+j)+1}\\&\quad \times \sum _{k=i}^{m-j}\left( {\begin{array}{c}m\\ k\end{array}}\right) \left( {\begin{array}{c}k\\ i\end{array}}\right) \left( {\begin{array}{c}m-k\\ j\end{array}}\right) \sum _{\ell =i}^{n-j} \left( {\begin{array}{c}n\\ \ell \end{array}}\right) \left( {\begin{array}{c}\ell \\ i\end{array}}\right) \left( {\begin{array}{c}n-\ell \\ j\end{array}}\right) . \end{aligned}$$

Since

$$\begin{aligned} \sum _{k=i}^{m-j}\left( {\begin{array}{c}m\\ k\end{array}}\right) \left( {\begin{array}{c}k\\ i\end{array}}\right) \left( {\begin{array}{c}m-k\\ j\end{array}}\right)&=\sum _{k=i}^{m-j}\left( {\begin{array}{c}m\\ i+j\end{array}}\right) \left( {\begin{array}{c}i+j\\ i\end{array}}\right) \left( {\begin{array}{c}m-(i+j)\\ k-i\end{array}}\right) \\&=2^{m-(i+j)}\left( {\begin{array}{c}m\\ i+j\end{array}}\right) \left( {\begin{array}{c}i+j\\ i\end{array}}\right) , \end{aligned}$$

we have

$$\begin{aligned} h^*({{\mathscr {A}}}_{K_{m+1,n+1}}, x)&=\sum _{i\ge 0}\sum _{j\ge 0}\left( {\begin{array}{c}i+j\\ i\end{array}}\right) ^{\!2}\left( {\begin{array}{c}m\\ i+j\end{array}}\right) \left( {\begin{array}{c}n\\ i+j\end{array}}\right) x^{i+j }(x+1)^{m+n-2(i+j)+1} \\&=\sum _{\alpha =0}^{\min (m, n)}\sum _{i=0}^\alpha \left( {\begin{array}{c}\alpha \\ i\end{array}}\right) ^{\!2}\left( {\begin{array}{c}m\\ \alpha \end{array}}\right) \left( {\begin{array}{c}n\\ \alpha \end{array}}\right) x^\alpha (x+1)^{m+n-2\alpha +1} \\&=\sum _{\alpha =0}^{\min (m, n)}\left( {\begin{array}{c}2 \alpha \\ \alpha \end{array}}\right) \left( {\begin{array}{c}m\\ \alpha \end{array}}\right) \left( {\begin{array}{c}n\\ \alpha \end{array}}\right) x^\alpha (x+1)^{m+n-2\alpha +1}. \end{aligned}$$

Finally, we conjecture the following:

Conjecture 5.11

The \(h^*\)-polynomial of any symmetric edge polytope of type A is \(\gamma \)-positive.

Twinned Chain Polytopes

In this section, we will apply Theorem 1.1 to twinned chain polytopes. For two lattice polytopes \({\mathscr {P}}, {\mathscr {Q}}\subset {\mathbb R}^d\), we set

$$\begin{aligned} \Gamma ({\mathscr {P}}, {\mathscr {Q}}):={\text {conv}}{({\mathscr {P}}\cup (- {\mathscr {Q}}))} \subset {\mathbb R}^d. \end{aligned}$$

Let P and Q be two finite posets on [d]. The twinned chain polytope of P and Q is the lattice polytope defined by \({\mathscr {C}}_{P,Q}:=\Gamma ({\mathscr {C}}_P,{\mathscr {C}}_Q)\). Then \({\mathscr {C}}_{P,Q}\) is reflexive. Moreover, \({\mathscr {C}}_{P,Q}\) has a flag, regular unimodular triangulation all of whose maximal simplices contain the origin [16, Prop. 1.2]. Hence we obtain

Corollary 6.1

Let P and Q be two finite posets on [d]. Then the \(h^*\)-polynomial of \({\mathscr {C}}_{P,Q}\) coincides with the h-polynomial of a flag triangulation of a sphere.

In [39, Prop. 2.2] it was shown that \({\mathscr {C}}_{P,Q}\) is locally anti-blocking. In general, for two finite posets \((P, <_P)\) and \((Q,<_Q)\) with \(P \cap Q = \emptyset \), the ordinal sum of P and Q is the poset \((P \oplus Q, <_{P \oplus Q})\) on \(P \oplus Q= P \cup Q\) such that \(i <_{P \oplus Q} j\) if and only if (a) \(i,j \in P\) and \(i <_P j\), or (b) \(i,j \in Q\) and \(i <_Q j\), or (c) \(i \in P\) and \(j \in Q\). Given a subset I of [d], we define the induced subposet of P on I to be the finite poset \((P_I,<_{P_I})\) on I such that \(i <_{P_I} j\) if and only if \(i <_P j\). For \(I \subset [d]\), let \({\overline{I}}:=[d] \setminus I\).

Proposition 6.2

[39, Prop. 2.2] Let P and Q be two finite posets on [d]. Then for each \(\varepsilon \in \{-1,1\}^d\), it follows that

$$\begin{aligned} {\mathscr {C}}_{P,Q} \cap {\mathbb R}^d_{\varepsilon }={\mathscr {C}}^{\pm }_{P_{I_\varepsilon } \oplus Q_{\overline{I_{\varepsilon }}}} \cap {\mathbb R}^d_{\varepsilon }, \end{aligned}$$

where \(I_{\varepsilon }=\{i \in [d] : \varepsilon _i=1 \}\).

From this result, Theorem 1.1, and Proposition 3.4 we obtain the following:

Theorem 6.3

Let P and Q be two finite posets on [d]. Then one has

$$\begin{aligned} h^*({\mathscr {C}}_{P,Q}, x)\,=\, \frac{1}{2^{d}}\sum _{\varepsilon \in \{-1,1\}^d}\!\!h^*({\mathscr {C}}^{(e)}_{R_{\varepsilon }},x)\,=\,(x+1)^d f_{P,Q}\biggl ( \frac{4x}{(x+1)^2} \biggr ), \end{aligned}$$

where \(I_{\varepsilon }=\{i \in [d] : \varepsilon _i=1 \}\) and \(R_{\varepsilon }\) is a naturally labeled poset that is obtained from \(P_{I_\varepsilon } \oplus Q_{{\overline{I}}_{\varepsilon }}\) by reordering the label and

$$\begin{aligned} f_{P,Q}(x)=\frac{1}{2^{d}}\sum _{\varepsilon \in \{-1,1\}^d}\!\!W^{(\ell )}_{R_{\varepsilon }}(x). \end{aligned}$$

In particular, \(h^*({\mathscr {C}}_{P,Q},x)\) is \(\gamma \)-positive. Moreover, \(h^*({\mathscr {C}}_{P,Q},x)\) is real-rooted if and only if \(f_{P,Q}(x)\) is real-rooted.

On the other hand, it is known that from \(h^*({\mathscr {C}}_{P,Q},x)\) we obtain \(h^*\)-polynomials of several non-locally anti-blocking lattice polytopes arising from the posets P and Q. The order polytope \({\mathscr {O}}_P\) [37] of P is the (0, 1)-polytope defined by

$$\begin{aligned} {\mathscr {O}}_P:=\{ {\mathbf{x}}\in [0,1]^d : x_i \le x_j\ \text {if}\ i <_P j \}. \end{aligned}$$

Given two lattice polytopes \({\mathscr {P}}, {\mathscr {Q}}\subset {\mathbb R}^d\), we define

$$\begin{aligned} {\mathscr {P}}*{\mathscr {Q}}:={\text {conv}}{ (({\mathscr {P}}\times \{0\}) \cup ({\mathscr {Q}}\times \{1\}) ) }\subset {\mathbb R}^{d+1}, \end{aligned}$$

which is called the Cayley sum of \({\mathscr {P}}\) and \({\mathscr {Q}}\), and define

$$\begin{aligned} \Omega ({\mathscr {P}},{\mathscr {Q}}):={\text {conv}}{(({\mathscr {P}}\times \{1\}) \cup (-{\mathscr {Q}}\times \{-1\}) )} \subset {\mathbb R}^{d+1}. \end{aligned}$$

Proposition 6.4

[16, Thm. 1.1] Let P and Q be two finite posets on [d]. Then

$$\begin{aligned}h^*({\mathscr {C}}_{P,Q},x)=h^*(\Gamma ({\mathscr {O}}_P,{\mathscr {C}}_Q),x).\end{aligned}$$

Furthermore, if P and Q have a common linear extension, then

$$\begin{aligned}h^*({\mathscr {C}}_{P,Q},x)=h^*(\Gamma ({\mathscr {O}}_P,{\mathscr {O}}_Q),x).\end{aligned}$$

Proposition 6.5

[18, Thm. 1.4] Let P and Q be two finite posets on [d]. Then

$$\begin{aligned}(1+x)h^*({\mathscr {C}}_{P,Q},x)=h^*(\Omega ({\mathscr {O}}_P,{\mathscr {C}}_Q),x).\end{aligned}$$

Furthermore, if P and Q have a common linear extension, then

$$\begin{aligned} (1+x)h^*({\mathscr {C}}_{P,Q},x)=h^*(\Omega ({\mathscr {O}}_P,{\mathscr {O}}_Q),x).\end{aligned}$$

Proposition 6.6

[17, Thm. 4.1] Let P and Q be two finite posets on [d]. Then

$$\begin{aligned}h^*({\mathscr {C}}_{P,Q},x)=h^*({\mathscr {O}}_P * {\mathscr {C}}_Q,x).\end{aligned}$$

From these propositions and Theorem 6.3, we obtain the following:

Corollary 6.7

Let P and Q be two finite posets on [d]. Then the \(h^*\)-polynomials of \(\Gamma ({\mathscr {O}}_P,{\mathscr {C}}_Q)\), \(\Omega ({\mathscr {O}}_P, {\mathscr {C}}_Q)\), \({\mathscr {O}}_P*{\mathscr {C}}_Q\), and \(\Omega ({\mathscr {C}}_P,{\mathscr {C}}_Q)\) are \(\gamma \)-positive. Furthermore, if P and Q have a common linear extension, then the \(h^*\)-polynomials of \(\Gamma ({\mathscr {O}}_P,{\mathscr {O}}_Q)\) and \(\Omega ({\mathscr {O}}_P,{\mathscr {O}}_Q)\) are also \(\gamma \)-positive.

In the rest of this section, we introduce enriched (PQ)-partitions and we show that the Ehrhart polynomial of \({\mathscr {C}}_{P,Q}\) coincides with a counting polynomial of enriched (PQ)-partitions. Assume that P and Q are naturally labeled. We say that a map \(f :[d] \rightarrow {\mathbb Z}\) is an enriched (P, Q)-partition if, for all \(x, y \in [d]\), it satisfies

  • \(x <_P y\), \(f(x) \ge 0\), and \(f(y) \ge 0 \Rightarrow f(x) \le f(y)\);

  • \(x <_Q y\), \(f(x) \le 0\), and \(f(y) \le 0 \Rightarrow f(x) \ge f(y)\).

For a map \(f :[d] \rightarrow {\mathbb Z}\), we set

$$\begin{aligned} m(f) = \min { \{ \{0\} \cup \{ f(x) : x \in [d] \} \}}\quad \text {and}\quad M(f) = \max {\{ \{0\} \cup f(x) : x \in [d]\} \}}. \end{aligned}$$

For each \(0 < m \in {\mathbb Z}\), let \(\Omega _{P,Q}^{(e)}(m)\) denote the number of enriched (PQ)-partitions \(f:[d] \rightarrow {\mathbb Z}\) with \(M(f) - m(f) \le m\).

Theorem 6.8

Let P and Q be two finite posets on [d]. Then one has

$$\begin{aligned} L_{{\mathscr {C}}_{P,Q}}(m)=\Omega _{P,Q}^{(e)}(m). \end{aligned}$$

Proof

Let F(m) stand for the set of enriched (PQ)-partitions with \(M(f)- m(f) \le m\). We show that there exists a bijection from \(m{\mathscr {C}}_{P,Q} \cap {\mathbb Z}^d\) to F(m). Take \(f \in F(m)\) and set \(m(f) = a\) and \(M(f)=b\). We set

$$\begin{aligned} I=\{i \in [d] : f(i) \ge 0 \}. \end{aligned}$$

Let

$$\begin{aligned} x_i={\left\{ \begin{array}{ll} f(i) &{}\quad \text {if }i \in I\text { is minimal in} \ P_I,\\ \min { \{ f(i) -f(j) : i\ \text {covers}\ j\ \text {in}\ P_I\}} &{}\quad \text {if }i \in I\text { is not minimal in }P_I,\\ -|f(i)| &{}\quad \text {if }i \in {\overline{I}}\text { is minimal in }Q_{{\overline{I}}},\\ -\min { \{ |f(i)| -|f(j)| : i\ \text {covers}\ j\ \text {in}\ Q_{{\overline{I}}}\}}&{}\quad \text {if }i\in {\overline{I}}\text { is not minimal in }Q_{{\overline{I}}}. \end{array}\right. } \end{aligned}$$

Assume that \(I=\{1,\ldots ,k \}\) and \({\overline{I}}=\{k+1,\ldots ,d\}\). Then we have \((x_1,\ldots ,x_k) \in b {\mathscr {C}}_{P_I}\) and \((x_{k+1},\ldots ,x_d) \in a{\mathscr {C}}_{Q_{{\overline{I}}}}\) by a result of Stanley [37, Thm. 3.2]. Hence one obtains \((x_1,\ldots ,x_d) \in b {\mathscr {C}}_{P_I} \oplus a {\mathscr {C}}_{Q_{{\overline{I}}}} \subset m{\mathscr {C}}_{P,Q}\), where \(b {\mathscr {C}}_{P_I} \oplus a {\mathscr {C}}_{Q_{{\overline{I}}}}\) is the free sum of \(b {\mathscr {C}}_{P_I}\) and \(a {\mathscr {C}}_{Q_{{\overline{I}}}}\). Similarly, in general, it follows that \((x_1,\ldots ,x_d) \in m{\mathscr {C}}_{P,Q}\). Therefore, the map \(\varphi :F(m) \rightarrow m {\mathscr {C}}_{P,Q} \cap {\mathbb Z}^d\), \(\varphi (f)=(x_1,\ldots ,x_d)\) for each \(f \in F(m)\), is well defined.

Take \((x_1,\ldots ,x_d) \in m{\mathscr {C}}_{P,Q} \cap {\mathbb Z}^d\). We set \(I= \{ i \in [d] : x_i \ge 0 \}\) and define a map \(f:[d] \rightarrow {\mathbb Z}\) by

$$\begin{aligned} f(i) ={\left\{ \begin{array}{ll} \max {\{x_{j_1} + \dots + x_{j_k} : j_1<_{P_I} \cdots<_{P_I} j_k =i \}} &{}\quad \text {if }i \in I,\\ -\max {\{ |x_{j_1}| + \dots + |x_{j_k}| : j_1<_{Q_{{\overline{I}}}} \cdots <_{Q_{{\overline{I}}}} j_k =i \}} &{}\quad \text {if }i \in {\overline{I}}. \end{array}\right. } \end{aligned}$$

Assume that \(I=\{1,\ldots ,k \}\) and \({\overline{I}}=\{k+1,\ldots ,d\}\). Then one has \((x_1,\ldots ,x_d) \in m({\mathscr {C}}_{P_I} \oplus (-{\mathscr {C}}_{Q_{{\overline{I}}}})) \cap {\mathbb Z}^d\). Moreover, for some integers a and b with \(a \le 0 \le b\) and \(b-a \le m\), it follows that \((x_1,\ldots ,x_k) \in b {\mathscr {C}}_{P_I}\) and \((x_{k+1},\ldots ,x_d) \in a {\mathscr {C}}_{Q_{{\overline{I}}}}\). We define \(f_1:I \rightarrow {\mathbb Z}\) by \(f_1(i)=f(i)\), and \(f_2:{\overline{I}} \rightarrow {\mathbb Z}\) by \(f_2(i)=-f(i)\). From [37, proof of Thm. 3.2], it follows that \(0 \le f_1(i) \le b\) for any \(i \in I\) and \(f_1(x) \le f_1(y)\) if \(x_{<_{P_I}} y\), and \(0 \ge f_2(i) \ge a\) for any \(i \in {\overline{I}}\) and \(f_2(x) \le f_2(y)\) if \(x_{<_{Q_{{\overline{I}}}}} y\). Therefore, \(f:[d] \rightarrow {\mathbb Z}\) is an enriched (PQ)-partition with \(M(f)-m(f) \le b - a \le m\), namely, \(f \in F(m)\). Similarly, in general, it follows that \(f \in F(m)\). Thus, the map \(\psi :m{\mathscr {C}}_{P,Q} \cap {\mathbb Z}^d \rightarrow F(m)\), \(\psi ({\mathbf{x}})(i)=f(i)\) for each \({\mathbf{x}}=(x_1,\ldots ,x_d) \in m{\mathscr {C}}_{P,Q} \cap {\mathbb Z}^d\), is well defined.

Finally, we show that \(\varphi \) is a bijection. However, this immediately follows by the above and the argument in [37, proof of Thm. 3.2]. \(\square \)

Since \({\mathscr {C}}_{P,Q}\) is reflexive, we obtain

Corollary 6.9

Let P and Q be two finite naturally labeled posets on [d]. Then \(\Omega ^{(e)}_{P,Q}(m)\) is a polynomial in m of degree d and one has

$$\begin{aligned} \Omega ^{(e)}_{P,Q}(m)=(-1)^d\Omega ^{(e)}_{P,Q}(-m-1). \end{aligned}$$

References

  1. Ardila, F., Beck, M., Hoşten, S., Pfeifle, J., Seashore, K.: Root polytopes and growth series of root lattices. SIAM J. Discrete Math. 25(1), 360–378 (2011)

    MathSciNet  Article  Google Scholar 

  2. Athanasiadis, Ch.A.: Gamma-positivity in combinatorics and geometry. Séminaire Lotharingien de Combinatoire 77, # B77i (2016–2018)

  3. Batyrev, V.V.: Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties. J. Algebr. Geom. 3(3), 493–535 (1994)

    MathSciNet  MATH  Google Scholar 

  4. Braun, B.: An Ehrhart series formula for reflexive polytopes. Electron. J. Comb. 13(1), # 15 (2006)

    MathSciNet  Article  Google Scholar 

  5. Bruns, W., Römer, T.: \(h\)-Vectors of Gorenstein polytopes. J. Comb. Theory Ser. A 114(1), 65–76 (2007)

    MathSciNet  Article  Google Scholar 

  6. Conway, J.H., Sloane, N.J.A.: Low-dimensional lattices. VII. Coordination sequences. Proc. R. Soc. Lond. Ser. A 453(1966), 2369–2389 (1997)

    MathSciNet  Article  Google Scholar 

  7. Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties. Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence (2011)

    Google Scholar 

  8. D’Alì, A., Delucchi, E., Michałek, M.: Many faces of symmetric edge polytopes (2019). arXiv:1910.05193

  9. Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173. Springer, Heidelberg (2010)

    Google Scholar 

  10. Ehrhart, E.: Polynomês Arithmétiques et Méthode des Polyèdres en Combinatorie International Series of Numerical Mathematics, vol. 35. Birkhäuser, Basel–Stuttgart (1977)

  11. Fulkerson, D.R.: Blocking and anti-blocking pairs of polyhedra. Math. Program. 1, 168–194 (1971)

    MathSciNet  Article  Google Scholar 

  12. Fulkerson, D.R.: Anti-blocking polyhedra. J. Comb. Theory Ser. B 12, 50–71 (1972)

    MathSciNet  Article  Google Scholar 

  13. Gal, Ś.R.: Real Root Conjecture fails for five- and higher-dimensional spheres. Discrete Comput. Geom. 34(2), 269–284 (2005)

    MathSciNet  Article  Google Scholar 

  14. Herzog, J., Hibi, T., Ohsugi, H.: Binomial Ideals. Graduate Texts in Mathematics, vol. 279. Springer, Cham (2018)

    Book  Google Scholar 

  15. Hibi, T.: Dual polytopes of rational convex polytopes. Combinatorica 12(2), 237–240 (1992)

    MathSciNet  Article  Google Scholar 

  16. Hibi, T., Matsuda, K., Tsuchiya, A.: Gorenstein Fano polytopes arising from order polytopes and chain polytopes. arXiv:1507.03221

  17. Hibi, T., Ohsugi, H., Tsuchiya, A.: Integer decomposition property for Cayley sums of order and stable set polytopes. Michigan Math. J. (2020). https://doi.org/10.1307/mmj/1585792887

    MathSciNet  Article  MATH  Google Scholar 

  18. Hibi, T., Tsuchiya, A.: Facets and volume of Gorenstein Fano polytopes. Math. Nachr. 290(16), 2619–2628 (2017)

    MathSciNet  Article  Google Scholar 

  19. Hibi, T., Tsuchiya, A.: Reflexive polytopes arising from perfect graphs. J. Comb. Theory Ser. A 157, 233–246 (2018)

    MathSciNet  Article  Google Scholar 

  20. Higashitani, A.: Smooth Fano polytopes arising from finite directed graphs. Kyoto J. Math. 55(3), 579–592 (2015)

    MathSciNet  Article  Google Scholar 

  21. Higashitani, A., Jochemko, K., Michałek, M.: Arithmetic aspects of symmetric edge polytopes. Mathematika 65(3), 763–784 (2019)

    MathSciNet  Article  Google Scholar 

  22. Kálmán, T.: A version of Tutte’s polynomial for hypergraphs. Adv. Math. 244, 823–873 (2013)

    MathSciNet  Article  Google Scholar 

  23. Kohl, F., Olsen, M., Sanyal, R.: Unconditional reflexive polytopes. Discrete Comput. Geom. (2020). https://doi.org/10.1007/s00454-020-00199-8

    MathSciNet  Article  MATH  Google Scholar 

  24. Kreuzer, M., Skarke, H.: Complete classification of reflexive polyhedra in four dimensions. Adv. Theor. Math. Phys. 4(6), 1209–1230 (2000)

    MathSciNet  Article  Google Scholar 

  25. Lagarias, J.C., Ziegler, G.M.: Bounds for lattice polytopes containing a fixed number of interior points in a sublattice. Can. J. Math. 43(5), 1022–1035 (1991)

    MathSciNet  Article  Google Scholar 

  26. Matsui, T., Higashitani, A., Nagazawa, Y., Ohsugi, H., Hibi, T.: Roots of Ehrhart polynomials arising from graphs. J. Algebr. Comb. 34(4), 721–749 (2011)

    MathSciNet  Article  Google Scholar 

  27. Nill, B.: Classification of pseudo-symmetric simplicial reflexive polytopes. In: Algebraic and Geometric Combinatorics (Anogia 2005). Contemp. Math., vol. 423, pp. 269–282. American Mathematical Society, Providence (2006)

  28. Ohsugi, H., Hibi, T.: Centrally symmetric configurations of integer matrices. Nagoya Math. J. 216, 153–170 (2014)

    MathSciNet  Article  Google Scholar 

  29. Ohsugi, H., Shibata, K.: Smooth Fano polytopes whose Ehrhart polynomial has a root with large real part. Discrete Comput. Geom. 47(3), 624–628 (2012)

    MathSciNet  Article  Google Scholar 

  30. Ohsugi, H., Tsuchiya, A.: Enriched chain polytopes. Israel J. Math. 237, 485–500 (2020)

    MathSciNet  Article  Google Scholar 

  31. Ohsugi, H., Tsuchiya, A.: Reflexive polytopes arising from bipartite graphs with \(\gamma \)-positivity associated to interior polynomials. Selecta Math. (N.S.), to appear

  32. O’Keeffe, M.: Coordination sequences for lattices. Zeitschrift für Kristallographie 210(12), 905–908 (1995)

    Article  Google Scholar 

  33. Petersen, T.K.: Enriched \(P\)-partitions and peak algebras. Adv. Math. 209(2), 561–610 (2007)

    MathSciNet  Article  Google Scholar 

  34. Petersen, T.K.: Eulerian Numbers. Birkhäuser Advanced Texts. Basler Lehrbücher. Birkhäuser, New York (2015)

    Book  Google Scholar 

  35. Schrijver, A.: Theory of Linear and Integer Programming. Wiley-Interscience Series in Discrete Mathematics. Wiley, Chichester (1986)

    Google Scholar 

  36. Stanley, R.P.: Decompositions of rational convex polytopes. Ann. Discrete Math. 6, 333–342 (1980)

    MathSciNet  Article  Google Scholar 

  37. Stanley, R.P.: Two poset polytopes. Discrete Comput. Geom. 1(1), 9–23 (1986)

    MathSciNet  Article  Google Scholar 

  38. Stembridge, J.R.: Enriched \(P\)-partitions. Trans. Am. Math. Soc. 349(2), 763–788 (1997)

    MathSciNet  Article  Google Scholar 

  39. Tsuchiya, A.: Volume, facets and dual polytopes of twinned chain polytopes. Ann. Comb. 22(4), 875–884 (2018)

    MathSciNet  Article  Google Scholar 

  40. Wang, C., Yu, J.: Toric \(h\)-vectors and Chow Betti numbers of dual hypersimplices (2017). arXiv:1707.04581

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Acknowledgements

The authors are grateful to the anonymous referees for their careful reading and helpful comments. The authors were partially supported by JSPS KAKENHI 18H01134, 19K14505, and 19J00312.

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Ohsugi, H., Tsuchiya, A. The \(h^*\)-Polynomials of Locally Anti-Blocking Lattice Polytopes and Their \(\gamma \)-Positivity. Discrete Comput Geom 66, 701–722 (2021). https://doi.org/10.1007/s00454-020-00236-6

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Keywords

  • Lattice polytope
  • Unconditional polytope
  • Anti-blocking polytope
  • Locally anti-blocking polytope
  • Reflexive polytope
  • \(h^*\)-polynomial
  • \(\gamma \)-positive

Mathematics Subject Classification

  • 05A15
  • 05C31
  • 13P10
  • 52B12
  • 52B20