Abstract
A lattice polytope \(\mathscr {P} \subset \mathbb {R}^d\) is called a locally antiblocking 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 antiblocking polytope by reflections of coordinate hyperplanes. We give a formula for the \(h^*\)polynomials of locally antiblocking lattice polytopes. In particular, we discuss the \(\gamma \)positivity of \(h^*\)polynomials of locally antiblocking 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 antiblocking 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 \({\mathbf{x}}\in {\mathscr {P}}\). Antiblocking 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 antiblocking lattice polytope \({\mathscr {P}}\subset {\mathbb R}_{\ge 0}^d\) of dimension d, we define
Since \({\mathscr {P}}\) is an antiblocking 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 antiblocking [23] if, for each \(\varepsilon \in \{1,1\}^d\), there exists an antiblocking 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 antiblocking.
In the present paper, we investigate the \(h^*\)polynomials of locally antiblocking lattice polytopes. First, we give a formula for the \(h^*\)polynomials of locally antiblocking lattice polytopes in terms of that of unconditional lattice polytopes.
Theorem 1.1
Let \({\mathscr {P}}\subset {\mathbb R}^d\) be a locally antiblocking lattice polytope of dimension d and for each \(\varepsilon \in \{1,1\}^d\), let \({\mathscr {P}}_{\varepsilon }\) be an antiblocking 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
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 antiblocking 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 antiblocking reflexive polytopes were characterized. In fact, a locally antiblocking 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 antiblocking 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 antiblocking 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 antiblocking 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 pseudodel Pezzo polytopes, we will show that the \(h^*\)polynomial of every pseudosymmetric 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 antiblocking 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 (P, Q)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 (P, Q)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 antiblocking polytopes and unconditional polytopes. In Sect. 4, we will investigate the \(h^*\)polynomials of locally antiblocking 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
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
is called the Ehrhart series of \({\mathscr {P}}\). It is well known that it can be expressed as a rational function of the form
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
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.

(RR)
We say that f is realrooted if all its roots are real.

(LC)
We say that f is logconcave if \(a_i^2 \ge a_{i1}a_{i+1}\) for all i.

(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
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 \(f(x)=\sum _{i \ge 0}\gamma _ix^i (1+x)^{d2i}\). 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 realrooted if and only if its \(\gamma \)polynomial is realrooted. If f is palindromic and realrooted, 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 hpolynomial of a flag triangulation of a sphere [5]. For the hpolynomial of a flag triangulation of a sphere, Gal [13] conjectured the following:
Conjecture 2.3
The hpolynomial of any flag triangulation of a sphere is \(\gamma \)positive.
Classes of AntiBlocking Polytopes and Unconditional Polytopes
In this section, we introduce several classes of antiblocking 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 antiblocking 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
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 antiblocking polytope.
Locally antiblocking 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 antiblocking 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 1element 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
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 antiblocking 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 Ppartitions. We assume that P is naturally labeled. A map \(f:P \rightarrow {\mathbb Z}\setminus \{0\}\) is called an enriched Ppartition [38] if, for all \(x, y \in P\) with \(x <_P y\), f satisfies
A map \(f:P \rightarrow {\mathbb Z}\) is called a left enriched Ppartition [33] if, for all \(x, y \in P\) with \(x <_P y\), f satisfies
The symbol \(\Omega _P^{(\ell )}(m)\) will denote the number of left enriched Ppartitions \(f:P \rightarrow {\mathbb Z}\) 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
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 \(1 \le i \le d1\) such that \(\pi _{i1} <\pi _i > \pi _{i+1} \), 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
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
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
Then \(B_G = {\mathscr {P}}_\Delta \) where \(\Delta \) is a simplicial complex on [d] obtained by regarding G as a 1dimensional 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 nonempty 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
Proposition 3.6
[31]Let G be a bipartite graph on [d]. Then the \(h^*\)polynomial of the reflexive polytope \({\mathscr {B}}_G\) is
In particular, \(h^*({\mathscr {B}}_G, x)\) is \(\gamma \)positive.
\(h^*\)Polynomials of Locally AntiBlocking Lattice Polytopes
In the present section, we prove Theorem 1.1, that is, a formula for the \(h^*\)polynomials of locally antiblocking lattice polytopes in terms of that of unconditional lattice polytopes. Given a subset \(J=\{j_1,\dots , j_r\}\) of [d], let
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 antiblocking lattice polytope. Then we have
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 \(d1\) 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
Hence the Ehrhart polynomial \(L_{{\mathscr {P}}^\pm }(m)\) satisfies the following:
Thus the Ehrhart series satisfies
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
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 \( \{ \varepsilon ' \in \{1,1\}^d : \pi _J(\varepsilon ') = \varepsilon \} = 2^{dr}\). Thus
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 antiblocking 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 antiblocking 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
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 \((d1)\)dimensional cross polytope. Hence we have \(h^*({\mathscr {A}}_G,x)= (x+1)^{d1}\).
It is known [26, Prop. 4.1] that the dimension of \({\mathscr {A}}_G\) is \(d1\) 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 realrooted 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 2connected if needed.
Proposition 5.2
Let G be a graph and let \(G_1,\ldots , G_s\) be 2connected components of G. Then the \(h^*\)polynomial of \({{\mathscr {A}}}_{G}\) satisfies
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
We now study the \(h^*\)polynomial of \({{\mathscr {A}}}_{{\widehat{G}}}\). Given a subset \(S \subset [d]\),
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 \({{\text {Cut}}(G)} = 2^{d1}\). 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 antiblocking reflexive polytope whose \(h^*\)polynomial is
where
In particular, \(h^*({{\mathscr {A}}}_{{\widehat{G}}}, x)\) is \(\gamma \)positive. Moreover, \(h^*({{\mathscr {A}}}_{{\widehat{G}}}, x)\) is realrooted if and only if \(f_G(x)\) is realrooted.
Proof
Let \({{\mathscr {P}}} \subset {\mathbb R}^d\) be the convex hull of
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
Hence \({{\mathscr {P}}} \cap {\mathbb R}_\varepsilon ^d = {\mathscr {B}}_{E_{S_\varepsilon }} \cap {\mathbb R}_\varepsilon ^d\). Thus \({\mathscr {P}}\) is a locally antiblocking polytope and
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

(i)
Delete e and identify the vertices i and j

(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
be the Laurent polynomial ring over a field K and let
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:
where C is an even cycle in G of length 2k with a fixed orientation, and I is a ksubset of C such that \(e_\ell \notin I\) for \(\ell =\min {\{i:e_i\in C\}}\);
where C is an odd cycle in G of length \(2k+1\) and I is a \((k+1)\)subset of C;
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
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:
where C is an even cycle in G of length 2k with a fixed orientation and \(e_1 \notin C\), and I is a ksubset of C such that \(e_\ell \notin I\) for \(\ell = \min {\{i : e_i \in C\}}\);
where \(C \cup \{e_1\}\) is an even cycle in G of length \(2k+2\) and I is a \((k+1)\)subset of C;
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
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:

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

(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].
PseudoSymmetric Simplicial Reflexive Polytopes
A lattice polytope \({\mathscr {P}}\subset {\mathbb R}^d\) is called pseudosymmetric 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 pseudosymmetric 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 \(V_{2m} = {\text {conv}}{( \pm {\mathbf{e}}_1,\dots ,\pm {\mathbf{e}}_{2m}, \pm ( {\mathbf{e}}_1 + \dots + {\mathbf{e}}_{2m}) )}\);

pseudodel 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
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 \(d1\). 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:
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
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
Moreover, if \(d = 2m +1\), \(2 \le m \in {\mathbb Z}\), then the coefficient of \(x^m\) in
is
and the other coefficient is arising from \((x+1) h^*({\mathscr {A}}_{C_{d1}},x)\). By a recursive formula in [29, Thm. 2.3], we have
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
\(\square \)
Thus it turns out that any pseudosymmetric 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 pseudosymmetric simplicial reflexive polytope is \(\gamma \)positive.
Proof
From results by Nill [27], any pseudosymmetric simplicial reflexive polytope is a free sum of cross polytopes, del Pezzo polytopes, and pseudodel 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 pseudosymmetric 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
If the edge set of \(H \in {\text {Cut}}(K_{d1})\) is \(E_S\) with \(S \subset [d1]\), then H is a complete bipartite graph \(K_{S, d1S}\) and
(Here \(K_{0,d1}\) denotes an empty graph.) It then follows that
Example 5.10
[21] Let \(G=K_{m,n}\). Then \({\widetilde{G}} = K_{m+1,n+1}\) and
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 \(K_{mk,n\ell }\), and hence
where \(k=V_1 \cap S\) and \(\ell = n V_2 \cap S\). It then follows that
Since
we have
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
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 hpolynomial of a flag triangulation of a sphere.
In [39, Prop. 2.2] it was shown that \({\mathscr {C}}_{P,Q}\) is locally antiblocking. 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
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
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
In particular, \(h^*({\mathscr {C}}_{P,Q},x)\) is \(\gamma \)positive. Moreover, \(h^*({\mathscr {C}}_{P,Q},x)\) is realrooted if and only if \(f_{P,Q}(x)\) is realrooted.
On the other hand, it is known that from \(h^*({\mathscr {C}}_{P,Q},x)\) we obtain \(h^*\)polynomials of several nonlocally antiblocking 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
Given two lattice polytopes \({\mathscr {P}}, {\mathscr {Q}}\subset {\mathbb R}^d\), we define
which is called the Cayley sum of \({\mathscr {P}}\) and \({\mathscr {Q}}\), and define
Proposition 6.4
[16, Thm. 1.1] Let P and Q be two finite posets on [d]. Then
Furthermore, if P and Q have a common linear extension, then
Proposition 6.5
[18, Thm. 1.4] Let P and Q be two finite posets on [d]. Then
Furthermore, if P and Q have a common linear extension, then
Proposition 6.6
[17, Thm. 4.1] Let P and Q be two finite posets on [d]. Then
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 (P, Q)partitions and we show that the Ehrhart polynomial of \({\mathscr {C}}_{P,Q}\) coincides with a counting polynomial of enriched (P, Q)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
For each \(0 < m \in {\mathbb Z}\), let \(\Omega _{P,Q}^{(e)}(m)\) denote the number of enriched (P, Q)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
Proof
Let F(m) stand for the set of enriched (P, Q)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
Let
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
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 \(ba \le m\), 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 (P, Q)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
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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 AntiBlocking Lattice Polytopes and Their \(\gamma \)Positivity. Discrete Comput Geom 66, 701–722 (2021). https://doi.org/10.1007/s00454020002366
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DOI: https://doi.org/10.1007/s00454020002366
Keywords
 Lattice polytope
 Unconditional polytope
 Antiblocking polytope
 Locally antiblocking polytope
 Reflexive polytope
 \(h^*\)polynomial
 \(\gamma \)positive
Mathematics Subject Classification
 05A15
 05C31
 13P10
 52B12
 52B20