The $h^{*}$-polynomial of the cut polytope of $K_{2,m}$ in the lattice spanned by its vertices

TL;DR

This paper explicitly computes the $h^{*}$-polynomial of the cut polytope for the complete bipartite graph $K_{2,m}$, advancing understanding of lattice polytope invariants in combinatorial optimization.

Contribution

It provides the first explicit formula for the $h^{*}$-polynomial of the cut polytope of $K_{2,m}$ using Gröbner bases, filling a gap beyond trees.

Findings

Explicit $h^{*}$-polynomial for $K_{2,m}$ cut polytope

Uses Gröbner bases of toric ideals

Advances classification of cut polytopes

Abstract

The cut polytope of a graph is an important object in several fields, such as functional analysis, combinatorial optimization, and probability. For example, Sturmfels and Sullivant showed that the toric ideals of cut polytopes are useful in algebraic statistics. In the theory of lattice polytopes, the $h^{*}$-polynomial is one of the most important invariants. The necessary and sufficient condition in terms of graphs that the $h^{*}$-polynomial of a cut polytope is palindromic is known. However, except for trees, there are no classes of graphs for which the $h^{*}$-polynomial of their cut polytope is explicitly specified. In the present paper, we determine the $h^{*}$-polynomial of the cut polytope of complete bipartite graph $K_{2,m}$ using the theory of Gr\"{o}bner bases of toric ideals.

Tables

Table 1. Table 1: Number of squarefree standard monomials

	$A_1=\varnothing$	$A_1\ne \varnothing$
$A_k=\{3,\dots ,n\}$	$\left(k-1\right)!\left\{\frac{n-2}{k-1}\right\}$	$k!\left\{\frac{n-2}{k}\right\}$
$A_k\ne \{3,\dots ,n\}$	$k!\left\{\frac{n-2}{k}\right\}$	$\left(k+1\right)!\left\{\frac{n-2}{k+1}\right\}$

Equations

$$\mathrm{conv}({\mathbf{v}}_1,\dots ,{\mathbf{v}}_l)=\left\{\sum _{i=1}^l{r}_i{\mathbf{v}}_i|0\le r_i\in \mathbb{R},\sum _{i=1}^l{r}_i=1\right\}.$$

$${\mathcal{A}}_{\mathcal{P}}=\left(\begin{array}{ccc}{\mathbf{a}}_1& \cdots & {\mathbf{a}}_n\\ 1& \cdots & 1\end{array}\right).$$

$$i(\mathcal{P},m)=|m{\mathcal{P}}^{\prime }\cap \mathbb{Z}{\mathcal{A}}_{\mathcal{P}}|,$$

$$\mathbb{Z}{\mathcal{A}}_{\mathcal{P}}=\mathbb{Z}\left(\begin{array}{c}{\mathbf{a}}_1\\ 1\end{array}\right)+\cdots +\mathbb{Z}\left(\begin{array}{c}{\mathbf{a}}_n\\ 1\end{array}\right).$$

$$1+\sum _{m=1}^{\infty }i(\mathcal{P},m)x^m=\frac{h^{\ast }(\mathcal{P},x)}{(1-x{)}^{d+1}}.$$

$$d_i=\{\begin{array}{ll}1& \text{∣A∩ei∣=1,}\\ 0& \text{otherwise.}\end{array}$$

$$\mathrm{Cut}(G)=\mathrm{conv}({\delta }_{A|B}|A,B\subset V(G),A\cap B=\varnothing ,A\cup B=V(G)).$$

$$A_n(x):=\sum _{w\in {\mathfrak{S}}_n}{x}^{\mathrm{des}(w)}$$

$$i(\mathcal{P},m)=|m{\mathcal{P}}^{\prime }\cap \mathbb{Z}{\mathcal{A}}_{\mathcal{P}}|$$

$$|m\mathcal{P}\cap {\mathbb{Z}}^d|$$

$$\mathrm{conv}(A)=\bigcup _{F\in \Delta }F.$$

$$\Delta ({\mathrm{in}}_{<}(I_A)):=\left\{\mathrm{conv}(B)|B\subset \{{\mathbf{a}}_1,\dots ,{\mathbf{a}}_n\},\prod _{{\mathbf{a}}_i\in B}{x}_i\notin \sqrt{{\mathrm{in}}_{<}(I_A)}\right\}.$$

$$A=({\mathbf{a}}_1,{\mathbf{a}}_2,{\mathbf{a}}_3,{\mathbf{a}}_4,{\mathbf{a}}_5)=\left(\begin{array}{ccccccccc}0& & 1& & 1& & 0& & 1\\ 0& & 1& & 0& & 1& & 1\\ 0& & 0& & 1& & 1& & 1\\ 1& & 1& & 1& & 1& & 1\end{array}\right).$$

$${\sigma }_1=\mathrm{conv}({\mathbf{a}}_1,{\mathbf{a}}_2,{\mathbf{a}}_3,{\mathbf{a}}_4)\text{mbox}and{\sigma }_2=\mathrm{conv}({\mathbf{a}}_2,{\mathbf{a}}_3,{\mathbf{a}}_4,{\mathbf{a}}_5).$$

$${\mathcal{A}}_{\mathrm{Cut}(G)}=\left(\begin{array}{ccccccc}{\delta }_{A_1|B_1}& & {\delta }_{A_2|B_2}& & \cdots & & {\delta }_{A_N|B_N}\\ & & & & & & \\ 1& & 1& & \cdots & & 1\end{array}\right),$$

$$K[q]=K[q_{A_1|B_1},\dots ,q_{A_N|B_N}],$$

$${\mathcal{A}}_{\mathrm{Cut}(K_{2,3})}=\text{setcounter}MaxMatrixCols16\left(\begin{array}{cccccccccccccccc}0& 0& 0& 0& 1& 1& 1& 1& 0& 0& 0& 0& 1& 1& 1& 1\\ 0& 0& 0& 0& 0& 0& 0& 0& 1& 1& 1& 1& 1& 1& 1& 1\\ 0& 0& 1& 1& 1& 1& 0& 0& 1& 1& 0& 0& 0& 0& 1& 1\\ 0& 0& 1& 1& 0& 0& 1& 1& 0& 0& 1& 1& 0& 0& 1& 1\\ 0& 1& 0& 1& 1& 0& 1& 0& 1& 0& 1& 0& 0& 1& 0& 1\\ 0& 1& 0& 1& 0& 1& 0& 1& 0& 1& 0& 1& 0& 1& 0& 1\\ 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1\end{array}\right),$$

$$h^{\ast }(\mathrm{Cut}(K_{2,3}),x)=x^5+9x^4+26x^3+26x^2+9x+1=(x+1)(x^2+4x+1{)}^2.$$

$$q_{A_1|\{1,2\}\cup B_1}\dots q_{A_k|\{1,2\}\cup B_k}{q}_{\{1\}\cup A_1^{\prime }|\{2\}\cup B_1^{\prime }}\dots q_{\{1\}\cup A_l^{\prime }|\{2\}\cup B_l^{\prime }}\in K[q]$$

$$(k-1)!\{\genfrac{}{}{0ex}{}{n-2}{k-1}\}+2k!\{\genfrac{}{}{0ex}{}{n-2}{k}\}+(k+1)!\left\{\genfrac{}{}{0ex}{}{n-2}{k+1}\right\},$$

$$2k!\left\{\genfrac{}{}{0ex}{}{n-2}{k}\right\}+(k+1)!\{\genfrac{}{}{0ex}{}{n-2}{k+1}\},$$

$$h_{\Delta }(x)=\sum _{i=0}^{d+1}{f}_{i-1}{x}^i(1-x{)}^{d+1-i}.$$

$$h^{\ast }(\mathcal{P},x)=h_{\Delta }(x)=(x+1)(A_{n-2}(x){)}^2,$$

$$A_n(x)=\sum _{k=1}^{n}k!\left\{\genfrac{}{}{0ex}{}{n}{k}\right\}(x-1{)}^{n-k}.$$

$$\left\{\genfrac{}{}{0ex}{}{n}{k}\right\}=\sum _{m=1}^k(-1{)}^{k-m}{m}^n(\genfrac{}{}{0ex}{}{k}{m}).$$

$$f_{\Delta }(x)=\sum _{k=0}^{2n-3}{f}_{k-1}{x}^k.$$

$$q_{A_1|\{1,2\}\cup B_1}\dots q_{A_{\alpha }|\{1,2\}\cup B_{\alpha }}{q}_{\{1\}\cup A_{\alpha +1}^{\prime }|\{2\}\cup B_{\alpha +1}^{\prime }}\dots q_{\{1\}\cup A_k^{\prime }|\{2\}\cup B_k^{\prime }}\in K[q]$$

$$B_k$$

$$C_k$$

$$f_{\Delta }(x)=\sum _{k=0}^{2n-3}\sum _{\alpha =0}^k{B}_{\alpha }{C}_{k-\alpha }{x}^k=\left(\sum _{k=0}^{n-1}{B}_k{x}^k\right)\left(\sum _{k=0}^{n-2}{C}_k{x}^k\right).$$

$$h_{\Delta }(x)$$

Full text

1 Introduction

Given integer vectors $\bm{v}_{1},\ldots,\bm{v}_{l}\in{\mathbb{Z}}^{d}$, let

[TABLE]

A set $\mathcal{P}\subset{\mathbb{R}}^{d}$ is called a lattice polytope if there exist $\bm{v}_{1},\ldots,\bm{v}_{l}\in{\mathbb{Z}}^{d}$ such that $\mathcal{P}={\rm conv}(\bm{v}_{1},\ldots,\bm{v}_{l})$. Let $\mathcal{P}\subset{\mathbb{R}}^{d}$ be a lattice polytope of dimension $d$, where $\mathcal{P}\cap{\mathbb{Z}}^{d}=\{\bm{a}_{1},\ldots,\bm{a}_{n}\}$. Let $\mathcal{A_{\mathcal{P}}}$ be the integer matrix

[TABLE]

The normalized Ehrhart polynomial $i(\mathcal{P},m)$ is given by the following equation:

[TABLE]

where $m\in\mathbb{N}$, $\mathcal{P}^{\prime}={\rm conv}(\binom{\bm{a}_{1}}{1},\ldots,\binom{\bm{a}_{n}}{1})$, $m\mathcal{P}^{\prime}=\{m\bm{a}\ |\ \bm{a}\in\mathcal{P}^{\prime}\}$, and

[TABLE]

In general, $i(\mathcal{P},m)$ satisfies the following fundamental properties [4]:

• •

$i(\mathcal{P},m)$ is a polynomial of degree $d$ in $m$;

• •

$i(\mathcal{P},0)=1$.

The $h^{*}$-polynomial $h^{*}(\mathcal{P},x)$ of $\mathcal{P}$ in the lattice ${\mathbb{Z}}\mathcal{A_{\mathcal{P}}}$ is defined by

[TABLE]

In general, $h^{*}(\mathcal{P},x)$ satisfies the following properties:

• •

$h^{*}(\mathcal{P},x)=\sum_{i=0}^{d}h_{i}^{*}x^{i}$, where each $h_{i}^{*}$ is a nonnegative integer [18];

• •

$i(\mathcal{P},m)=\sum_{i=0}^{d}h_{i}^{*}\binom{m+d-i}{d}$;

• •

If $h_{d}^{*}>0$, then we have $h_{i}^{*}\geq h_{1}^{*}$ $(1\leq{i}\leq{d-1})$ [8, Theorem 1.1].

The third property is Hibi’s lower bound theorem. Since the $h^{*}$-polynomial is defined in the lattice ${\mathbb{Z}}\mathcal{A_{\mathcal{P}}}$, ${\mathcal{P}}$ is a spanning lattice polytope [10], and a generalization of Hibi’s lower bound theorem is known:

• •

$h_{i}^{*}\geq h_{1}^{*}$ $(1\leq{i}\leq\deg(h^{*}(\mathcal{P},x))-1)$ [9, Corollary 1.6].

A polynomial $f(x)$ of degree $s$ is said to be palindromic if $f(x)=x^{s}f(x^{-1})$. Let $K[\mathcal{A_{\mathcal{P}}}]$ be the toric ring of $\mathcal{P}$. (The toric ring will be defined in Section $1$.) If $K[\mathcal{A_{\mathcal{P}}}]$ is normal (i.e., ${\mathbb{Z}}_{\geq 0}\mathcal{A_{\mathcal{P}}}={\mathbb{Z}}\mathcal{A_{\mathcal{P}}}\cap{\mathbb{Q}}_{\geq 0}\mathcal{A_{\mathcal{P}}}$, see [21, Proposition 13.5]) and Gorenstein, then it is known by [7, Lemma $4.22\ (b)$] and [3, P.235] that the $h^{*}$-polynomial of ${\mathcal{P}}$ is palindromic. In the theory of lattice polytopes, $h^{*}$-polynomials are important objects to study. For example, the $h^{*}$-polynomials of stable set polytopes of graphs, order polytopes, and chain polytopes of posets were studied in [1, 19]. Let $G$ be a finite connected simple graph with vertex set $V(G)=\{1,2,\ldots,m\}$ and edge set $E(G)=\{e_{1},e_{2},\ldots,e_{r}\}$. For two subsets $A$ and $B$ of $V(G)$ such that $A\cap B=\emptyset,A\cup B=V(G)$, we define a vector $\delta_{A|B}=(d_{1},d_{2},\ldots,d_{r})\in\{0,1\}^{r}$ by

[TABLE]

Note that $\delta_{A|B}=\delta_{B|A}$. The cut polytope of $G$ is the $0/1$ polytope

[TABLE]

Example 1.1**.**

Let $G$ be a cycle of length $4$, where $V(G)=\{1,2,3,4\}$ and $E(G)=\{e_{1}=\{1,2\},e_{2}=\{2,3\},e_{3}=\{3,4\},e_{4}=\{1,4\}\}$. For subset $A=\{1,2\}\subset V(G)$, we compute $\delta_{A|B}=(d_{1},d_{2},d_{3},d_{4})$, where $B=\{3,4\}$. Since $|A\cap e_{1}|=2$, we have $d_{1}=0$. Similarly, we obtain $d_{2}=1,d_{3}=0$, and $d_{4}=1$. Hence, $\delta_{A|B}=(0,1,0,1)$. By computing $\delta_{A|B}$ for all subsets $A\subset V(G)$, we obtain the cut polytope ${\rm Cut}(G)={\rm conv}((0,0,0,0),(1,1,0,0),(1,0,1,0),(1,0,0,1),$

$(0,1,1,0),(0,1,0,1),(0,0,1,1),(1,1,1,1)).$

We define the graph theoretical terminology used in the present paper. A bridge of a graph $G$ is an edge of $G$ whose deletion increases the number of connected components, and a graph $G$ is said to be bridgeless if $G$ has no bridges. An induced cycle of $G$ is a cycle of $G$ that is an induced subgraph of $G$. A graph $G$ is said to be chordal if $G$ has no induced cycles of length $\geq 4$. A graph $H$ is called a minor of a graph $G$ if $H$ is obtained from $G$ by a sequence of contractions and deletions of edges. On the other hand, if we cannot obtain $H$ as a minor of $G$, then $G$ is said to be $H$-minor free. A complete graph with $n$ vertices is denoted by $K_{n}$, a complete bipartite graph with $m+n$ vertices is denoted by $K_{m,n}$, and a cycle of length $n$ is denoted by $C_{n}$. There is only one class of cut polytopes for which the $h^{*}$-polynomial is explicitly known. Nagel and Petrović [12] showed that, if $G$ is a tree with $n\geq 1$ edges, then the $h^{*}$-polynomial of the cut polytope in the lattice ${\mathbb{Z}}{\mathcal{A}}_{{\rm Cut}(G)}$ is the Eulerian polynomial

[TABLE]

of degree $n-1$. Here, $\mathfrak{S}_{n}$ is a symmetric group and ${\rm des}(w)=|\{i\ |\ w_{i}>w_{i+1}\}|$ for $w=w_{1}w_{2}\cdots w_{n}\in\mathfrak{S}_{n}$. It is known that $A_{n}(x)$ is palindromic and unimodal. Ohsugi [14] showed that the toric ring of the cut polytope ${\rm Cut}(G)$ of a graph $G$ is normal and Gorenstein if and only if $G$ is $K_{5}$-minor free and satisfies one of the following:

$G$ is a bipartite graph with no induced cycle of length $\geq 6$. 2. 2.

$G$ is a bridgeless chordal graph.

Thus, if $G$ satisfies one of the above conditions, then the $h^{*}$-polynomial of the cut polytope of $G$ is palindromic, since the toric ring is normal and Gorenstein.

In the present paper, we determine the $h^{*}$-polynomial of the cut polytope of a complete bipartite graph $K_{2,n-2}$ and show that the $h^{*}$-polynomial is $(x+1)(A_{n-2}(x))^{2}$ using the theory of Gröbner bases of toric ideals. See [7, 21] for the details of Gröbner bases and toric ideals.

Remark 1.2**.**

We discuss the normalized Ehrhart polynomial

[TABLE]

instead of the ordinary Ehrhart polynomial

[TABLE]

because the lattice spanned by $\delta_{A|B}$’s is important in the study of the cut polytopes. In fact, the characterization of the graph $G$ satisfying ${\mathbb{Z}}_{\geq 0}B_{G}={\mathbb{Z}}B_{G}\cap{\mathbb{Q}}_{\geq 0}B_{G}$, where $B_{G}={\rm Cut}(G)\cap{\mathbb{Z}}^{d}$, is an important open problem. See [6] and the references therein. In addition, it has been conjectured that ${\mathbb{Z}}_{\geq 0}\mathcal{A}_{{\rm Cut}(G)}={\mathbb{Z}}\mathcal{A}_{{\rm Cut}(G)}\cap{\mathbb{Q}}_{\geq 0}\mathcal{A}_{{\rm Cut}(G)}$ if and only if $G$ is $K_{5}$-minor free [22]. See [13, 22].

2 Standard monomials of cut ideals of $K_{2,n-2}$

Let $K[\bm{x}]=K[x_{1},\ldots,x_{n}]$ be a polynomial ring in $n$ variables over a field $K$. Let $\mathcal{M}_{n}$ be the set of all monomials of $K[\bm{x}]$. A total order $<$ on $\mathcal{M}_{n}$ is called a monomial order if $<$ satisfies the following properties:

• •

For all $1\neq u\in\mathcal{M}_{n}$, $1<u.$

• •

If $u<v$ ($u,v\in\mathcal{M}_{n}$), then $w\cdot u<w\cdot v$ for all $w\in\mathcal{M}_{n}$.

We fix a monomial order $<$. For a nonzero polynomial $f$ which belongs to $K[\bm{x}]$, the initial monomial in${}_{<}(f)$ of $f$ is the greatest monomial in $f$ with respect to $<$. The initial ideal of an ideal $I\subset K[\bm{x}]$ with respect to $<$ is defined by in${}_{<}(I)=\langle$ in${}_{<}(f)\ |\ 0\neq f\in I\rangle$. A finite subset $G=\{g_{1},\ldots,g_{s}\}\subset I$ is called a Gröbner basis of $I$ with respect to $<$ when in${}_{<}(I)=\langle$in${}_{<}(g_{1}),\ldots,$in${}_{<}(g_{s})\rangle$. See [7, Chapter 1] for the basic theory of Gröbner bases.

A $d\times n$ integer matrix $A=(\bm{a}_{1},\bm{a}_{2},\ldots,\bm{a}_{n})$ is called a configuration if there exists a vector $\bm{c}\in\mathbb{R}^{d}$ such that for all $1\leq{i}\leq{n}$, the inner product $\bm{a}_{i}\cdot\bm{c}$ is equal to $1$. Let $K[t_{1}^{\pm 1},t_{2}^{\pm 1},\ldots,t_{d}^{\pm 1}]$ be a Laurent polynomial ring over $K$. For an integer vector $\bm{b}=(b_{1},b_{2},\ldots,b_{d})\in\mathbb{Z}^{d}$, we define the Laurent monomial $\bm{t^{b}}=t_{1}^{b_{1}}t_{2}^{b_{2}}\cdots t_{d}^{b_{d}}\in K[t_{1}^{\pm 1},t_{2}^{\pm 1},\ldots,t_{d}^{\pm 1}]$ and the toric ring $K[A]=K[\bm{t}^{\bm{a}_{1}},\bm{t}^{\bm{a}_{2}},\ldots,\bm{t}^{\bm{a}_{n}}]$. Let $\pi$ be a homomorphism $\pi:K[\bm{x}]\rightarrow K[A]$, where $\pi(x_{i})=\bm{t}^{\bm{a}_{i}}$. The kernel of $\pi$ is called the toric ideal of $A$ and denoted by $I_{A}$. We often regard $A=(\bm{a}_{1},\ldots,\bm{a}_{n})$ as a set $A=\{\bm{a}_{1},\ldots,\bm{a}_{n}\}$. Suppose that a set $\Delta$ consists of simplices and that each vertex of $\sigma\in\Delta$ belongs to $A$. Then, $\Delta$ is called a covering of ${\rm conv}(A)$ if

[TABLE]

We say that a covering $\Delta$ is a triangulation of ${\rm conv}(A)$ if $\Delta$ is a simplicial complex. A triangulation $\Delta$ of a polytope ${\rm conv}(A)$ is unimodular if the normalized volume of any maximal simplex is equal to $1$. For a configuration $A$, the initial complex with respect to $<$ is defined by

[TABLE]

It is known that $\Delta({\rm in}_{<}(I_{A}))$ is a triangulation of conv$(A)$. Moreover, $\Delta({\rm in}_{<}(I_{A}))$ is unimodular if and only if ${\rm in}_{<}(I_{A})$ is generated by squarefree monomials.

Example 2.1**.**

Let $A$ be a configuration

[TABLE]

The toric ideal of $A$ is given by $I_{A}=\langle x_{1}x_{5}^{2}-x_{2}x_{3}x_{4}\rangle$. Let $<$ be the lexicographic order on $K[x_{1},x_{2},x_{3},x_{4},x_{5}]$ induced by the ordering $x_{1}>x_{2}>x_{3}>x_{4}>x_{5}$. Then, ${\rm in}_{<}(x_{1}x_{5}^{2}-x_{2}x_{3}x_{4})=x_{1}x_{5}^{2}$, ${\rm in}_{<}(I_{A})=\langle x_{1}x_{5}^{2}\rangle$ and $\sqrt{{\rm in}_{<}(I_{A})}=\langle x_{1}x_{5}\rangle$. Thus, the maximal simplices of $\Delta({\rm in}_{<}(I_{A}))$ are

[TABLE]

The triangulation $\Delta({\rm in}_{<}(I_{A}))$ is not unimodular, since the normalized volume of $\sigma_{1}$ is 2.

A monomial is said to be a standard monomial of a toric ideal $I_{A}$ with respect to a monomial order $<$ if the monomial does not belong to in${}_{<}(I_{A})$. If $\Delta({\rm in}_{<}(I_{A}))$ is unimodular, then the number of squarefree standard monomials of degree $i$ corresponds to the number of $(i-1)$-dimensional faces of $\Delta({\rm in}_{<}(I_{A}))$. See [7, 21] for details.

Let $G$ be a graph with $m$ vertices. We consider the configuration

[TABLE]

where $A_{i}\cap B_{i}=\emptyset,A_{i}\cup B_{i}=V(G)$ for $1\leq{i}\leq{N}$ and $N=2^{m-1}$. The toric ideal of $\mathcal{A}_{{\rm Cut}(G)}$ is called the cut ideal of $G$ and is denoted by $I_{G}$. The notion of cut ideals was introduced in [22]. The toric ring and ideal of $\mathcal{A}_{{\rm Cut}(G)}$ were investigated in, e.g., [5, 12, 11, 13, 16, 17]. The cut ideal $I_{G}$ of a graph $G$ is generated by quadratic binomials if and only if $G$ is $K_{4}$-minor free [5]. The cut ideal $I_{G}$ of a graph $G$ has a quadratic Gröbner basis if $G$ satisfies one of the following:

• •

$G$ is $(K_{4},C_{5})$-minor free, where $K_{4}$ is a complete graph with $4$ vertices, and $C_{5}$ is a cycle of length $5$ [17, Corollary 2.4];

• •

$G$ is a cycle of length $\leq 7$ [16, Theorem 2.3].

An unordered partition $A|B$ of the vertex set $V(G)$ consists of subsets $A,B\subset V(G)$ such that $A\cap B=\emptyset,A\cup B=V(G)$. Given an unordered partition $A|B$, we associate a variable $q_{A|B}$. In particular, $q_{A|B}=q_{B|A}$. Let $K[q]$ be the polynomial ring in $N=2^{m-1}$ variables over a field $K$ defined by

[TABLE]

where $\{A_{1}|B_{1},\ldots,A_{N}|B_{N}\}$ is the set of all unordered partitions of $V(G)$. Let $<$ be a reverse lexicographic order [7, Example $1.8(b)$] on $K[q]$ that satisfies $q_{A|B}<q_{C|D}$ with $\min\{|A|,|B|\}<\min\{|C|,|D|\}$. A quadratic Gröbner basis of the cut ideal of $K_{2,n-2}$ with respect to $<$ is given by the following proposition.

Proposition 2.2** ([17, Theorem 2.3]).**

Let $K_{2,n-2}$ be the complete bipartite graph on the vertex set $\{1,2\}\cup\{3,\ldots,n\}$ for $n\geq 4$. Then, a Gröbner basis of $I_{K_{2,n-2}}$ with respect to $<$ consists of

$q_{\{1\}\cup A|\{2\}\cup B}q_{\{1\}\cup B|\{2\}\cup A}-q_{\emptyset|[n]}q_{\{1,2\}|\{3,\ldots,n\}}$, 2. 2.

$q_{A|B}q_{C|D}-q_{{A\cap C}|{B\cup D}}q_{{A\cup C}|{B\cap D}}\ (1\in A\cap C,2\in B\cap D,A\not\subset C,C\not\subset A)$, 3. 3.

$q_{A|B}q_{C|D}-q_{{A\cap C}|{B\cup D}}q_{{A\cup C}|{B\cap D}}\ (1,2\in A\cap C,A\not\subset C,C\not\subset A)$.

The initial monomial of each binomial is the first monomial.

Example 2.3**.**

Let $K_{2,3}$ be the complete bipartite graph on the vertex set $\{1,2\}\cup\{3,4,5\}$. Then, the configuration $\mathcal{A}_{{\rm Cut}(K_{2,3})}$ is

[TABLE]

and a Gröbner basis of $I_{K_{2,3}}$ with respect to the monomial order $<$ consists of the following binomials:

$q_{\{1\}|\{2,3,4,5\}}q_{\{2\}|\{1,3,4,5\}}-q_{\emptyset|\{1,2,3,4,5\}}q_{\{1,2\}|\{3,4,5\}}$,

$q_{\{1,3\}|\{2,4,5\}}q_{\{2,3\}|\{1,4,5\}}-q_{\emptyset|\{1,2,3,4,5\}}q_{\{1,2\}|\{3,4,5\}}$,

$q_{\{1,4\}|\{2,3,5\}}q_{\{2,4\}|\{1,3,5\}}-q_{\emptyset|\{1,2,3,4,5\}}q_{\{1,2\}|\{3,4,5\}}$,

$q_{\{1,5\}|\{2,3,4\}}q_{\{2,5\}|\{1,3,4\}}-q_{\emptyset|\{1,2,3,4,5\}}q_{\{1,2\}|\{3,4,5\}}$,

$q_{\{3\}|\{1,2,4,5\}}q_{\{4,5\}|\{1,2,3\}}-q_{\emptyset|\{1,2,3,4,5\}}q_{\{1,2\}|\{3,4,5\}}$,

$q_{\{5\}|\{1,2,3,4\}}q_{\{3,4\}|\{1,2,5\}}-q_{\emptyset|\{1,2,3,4,5\}}q_{\{1,2\}|\{3,4,5\}}$,

$q_{\{4\}|\{1,2,3,5\}}q_{\{3,5\}|\{1,2,4\}}-q_{\emptyset|\{1,2,3,4,5\}}q_{\{1,2\}|\{3,4,5\}}$,

$q_{\{4\}|\{1,2,3,5\}}q_{\{5\}|\{1,2,3,4\}}-q_{\emptyset|\{1,2,3,4,5\}}q_{\{4,5\}|\{1,2,3\}},$

$q_{\{3\}|\{1,2,4,5\}}q_{\{5\}|\{1,2,3,4\}}-q_{\emptyset|\{1,2,3,4,5\}}q_{\{3,5\}|\{1,2,4\}},$

$q_{\{3\}|\{1,2,4,5\}}q_{\{4\}|\{1,2,3,5\}}-q_{\emptyset|\{1,2,3,4,5\}}q_{\{3,4\}|\{1,2,5\}},$

$q_{\{3,5\}|\{1,2,4\}}q_{\{4,5\}|\{1,2,3\}}-q_{\{5\}|\{1,2,3,4\}}q_{\{1,2\}|\{3,4,5\}},$

$q_{\{3,4\}|\{1,2,5\}}q_{\{3,5\}|\{1,2,4\}}-q_{\{3\}|\{1,2,4,5\}}q_{\{1,2\}|\{3,4,5\}},$

$q_{\{3,4\}|\{1,2,5\}}q_{\{4,5\}|\{1,2,3\}}-q_{\{4\}|\{1,2,3,5\}}q_{\{1,2\}|\{3,4,5\}}$

$q_{\{1,4\}|\{2,3,5\}}q_{\{1,5\}|\{2,3,4\}}-q_{\{1\}|\{2,3,4,5\}}q_{\{2,3\}|\{1,4,5\}},$

$q_{\{1,3\}|\{2,4,5\}}q_{\{1,5\}|\{2,3,4\}}-q_{\{1\}|\{2,3,4,5\}}q_{\{2,4\}|\{1,3,5\}},$

$q_{\{2,3\}|\{1,4,5\}}q_{\{2,4\}|\{1,3,5\}}-q_{\{2\}|\{1,3,4,5\}}q_{\{1,5\}|\{2,3,4\}},$

$q_{\{1,3\}|\{2,4,5\}}q_{\{1,4\}|\{2,3,5\}}-q_{\{1\}|\{2,3,4,5\}}q_{\{2,5\}|\{1,3,4\}},$

$q_{\{2,3\}|\{1,4,5\}}q_{\{2,5\}|\{1,3,4\}}-q_{\{2\}|\{1,3,4,5\}}q_{\{1,4\}|\{2,3,5\}},$

$q_{\{2,4\}|\{1,3,5\}}q_{\{2,5\}|\{1,3,4\}}-q_{\{2\}|\{1,3,4,5\}}q_{\{1,3\}|\{2,4,5\}}$.

The $h^{*}$-polynomial $h^{*}({\rm Cut}(K_{2,3}),x)$ is

[TABLE]

Since the dimension of the cut polytopes of $K_{2,n-2}$ is $2n-4$, the maximum degree of squarefree standard monomials is $2n-3$. Note that the initial monomial $q_{A|B}q_{C|D}$ of the binomials in the Gröbner basis in Proposition 2.2 satisfies one of the following conditions:

• •

$1\in A\cap C$ and $2\in B\cap D$,

• •

$1,2\in A\cap C$.

Hence, a squarefree monomial

[TABLE]

is standard if and only if both $q_{A_{1}|\{1,2\}\cup B_{1}}\ldots q_{A_{k}|\{1,2\}\cup B_{k}}$ and $q_{\{1\}\cup A_{1}^{\prime}|\{2\}\cup B_{1}^{\prime}}\ldots q_{\{1\}\cup A_{l}^{\prime}|\{2\}\cup B_{l}^{\prime}}$ are standard.

Proposition 2.4**.**

Each of the squarefree monomials

1. (1)

$q_{A_{1}|\{1,2\}\cup B_{1}}\ldots q_{A_{k}|\{1,2\}\cup B_{k}}$** 2. (2)

$q_{\{1\}\cup A_{1}^{\prime}|\{2\}\cup B_{1}^{\prime}}\ldots q_{\{1\}\cup A_{k}^{\prime}|\{2\}\cup B_{k}^{\prime}}$,

of degree $k\leq 2n-3$ is not divisible by the initial monomials if and only if, by changing indices if necessary,

1. (1)

$A_{1}\subsetneq A_{2}\subsetneq\dots\subsetneq A_{k}$, 2. (2)

$A^{\prime}_{1}\subsetneq A^{\prime}_{2}\subsetneq\dots\subsetneq A^{\prime}_{k}\ {\rm and}\ (A^{\prime}_{1},A^{\prime}_{k})\neq(\emptyset,\{3,\ldots,n\})$,

respectively. Moreover, the numbers of squarefree standard monomials of types $(1)$ and $(2)$ above are

[TABLE]

respectively, where $\genfrac{\{}{\}}{0.0pt}{}{n}{k}$ is the Stirling number of the second kind.

Proof.

Let $m_{1}=q_{A_{1}|\{1,2\}\cup B_{1}}\,\ldots\,q_{A_{k}|\{1,2\}\cup B_{k}}$ and $m_{2}=q_{\{1\}\cup A_{1}^{\prime}|\{2\}\cup B_{1}^{\prime}}\,\ldots\,q_{\{1\}\cup A_{k}^{\prime}|\{2\}\cup B_{k}^{\prime}}$ be squarefree monomials.

First, suppose that $m_{1}$ and $m_{2}$ are standard. Since $m_{1}$ is squarefree and is not divisible by the initial monomials $q_{A|B}q_{C|D}$ ($1,2\in A\cap C,A\not\subset C,C\not\subset A$), we can obtain $B_{1}\supsetneq\cdots\supsetneq B_{k-1}\supsetneq B_{k}$ by changing indices if necessary. Then, $A_{1}\subsetneq\cdots\subsetneq A_{k-1}\subsetneq A_{k}$. Similarly, since $m_{2}$ is squarefree and is not divisible by the initial monomials $q_{A|B}q_{C|D}$ ($1\in A\cap C,2\in B\cap D,A\not\subset C,C\not\subset A$), we can obtain $A_{1}^{\prime}\subsetneq\cdots\subsetneq A_{k-1}^{\prime}\subsetneq A_{k}^{\prime}$ by changing indices if necessary. Moreover, since $m_{2}$ is not divisible by the initial monomial $q_{\{1\}|\{2,3,\ldots,n\}}q_{\{1,3,\ldots,n\}|\{2\}}$ of the binomial $q_{\{1\}|\{2,3,\ldots,n\}}q_{\{1,3,\ldots,n\}|\{2\}}-q_{\emptyset|[n]}q_{\{1,2\}|\{3,\ldots,n\}}$, we have $(A^{\prime}_{1},A^{\prime}_{k})\neq(\emptyset,\{3,\ldots,n\})$.

Contrarily, suppose that $m_{1}$ and $m_{2}$ satisfy $A_{1}\subsetneq A_{2}\subsetneq\dots\subsetneq A_{k}$, $A^{\prime}_{1}\subsetneq A^{\prime}_{2}\subsetneq\dots\subsetneq A^{\prime}_{k}\ {\rm and}\ (A^{\prime}_{1},A^{\prime}_{k})\neq(\emptyset,\{3,\ldots,n\})$. Then, $m_{1}$ and $m_{2}$ are not divisible by the initial monomials $q_{A|B}q_{C|D}$ ($1,2\in A\cap C,A\not\subset C,C\not\subset A$) and $q_{A|B}q_{C|D}$ ($1\in A\cap C,2\in B\cap D,A\not\subset C,C\not\subset A$). Hence, $m_{1}$ is standard. Suppose that $m_{2}$ is not standard. Then, $m_{2}$ is divisible by the initial monomial $q_{\{1\}\cup A|\{2\}\cup B}q_{\{1\}\cup B|\{2\}\cup A}$. Since $A^{\prime}_{1}\subsetneq A^{\prime}_{2}\subsetneq\dots\subsetneq A^{\prime}_{k}$. Thus, we may assume that $A\subsetneq B$. Since $\{1\}\cup A|\{2\}\cup B$ is a partition, $A\cup B=\{3,\ldots,n\}$. Therefore, $(A,B)=(\emptyset,\{3,\ldots,n\})$. This contradicts the hypothesis $(A^{\prime}_{1},A^{\prime}_{k})\neq(\emptyset,\{3,\ldots,n\})$. Hence, $m_{2}$ is standard.

The Stirling number of the second kind $\genfrac{\{}{\}}{0.0pt}{}{a}{b}$ represents the number of ways to partition a set of $a$ objects into $b$ nonempty subsets. We obtain Table 1 by considering four cases for the number of squarefree standard monomials of type (1).

There is a restriction that $(A^{\prime}_{1},A^{\prime}_{k})\neq(\emptyset,\{3,\ldots,n\})$ for type $(2)$. Therefore, we obtain the desired number of squarefree standard monomials in each condition. ∎

3 $h^{*}$-polynomial of the cut polytope of $K_{2,n-2}$

Let $\Delta$ be a triangulation of a lattice polytope $\mathcal{P}$. The $f$-polynomial $f_{\Delta}(x)=\sum_{i=0}^{d+1}f_{i-1}x^{i}$ of $\Delta$ encodes the number $f_{i}$ of $i$-faces for $i=0,1\dots,d$ and $f_{-1}=1$. The $h$-polynomial $h_{\Delta}(x)=\sum_{i=0}^{d+1}h_{i}x^{i}$ is given by the following relation [2, P.185]:

[TABLE]

The following is known for $h^{*}$-polynomials and $h$-polynomials [2, Theorem $10.3$].

Proposition 3.1**.**

If $\mathcal{P}$ is a d-dimensional lattice polytope that admits a unimodular triangulation $\Delta$, then $\displaystyle h^{*}(\mathcal{P},x)=h_{\Delta}(x).$

We have the following theorem for the $h^{*}$-polynomial of the cut polytope of $K_{2,n-2}$.

Theorem 3.2**.**

Let ${\mathcal{P}}={\rm Cut}(K_{2,n-2})$ be the cut polytope of $K_{2,n-2}$, and let $\Delta$ be the unimodular triangulation $\Delta({\rm in}_{<}(I_{\mathcal{A}_{\mathcal{P}}}))$ of ${\mathcal{P}}$ with respect to the monomial order $<$ in Proposition 2.2. Then, the $h^{*}$-polynomial of ${\mathcal{P}}$ and the $h$-polynomial of $\Delta$ are

[TABLE]

where $A_{n}(x)$ is the Eulerian polynomial of degree $n-1$. In particular, the normalized volume of $\mathcal{P}$ is $h^{*}(\mathcal{P},1)=2((n-2)!)^{2}$.

Proof.

The Eulerian polynomial $A_{n}(x)$ satisfies the following condition [15, Theorem 9.1]:

[TABLE]

In addition, $\genfrac{\{}{\}}{0.0pt}{}{n}{k}$ is given by the following equation [20]:

[TABLE]

The $f$-polynomial of $\Delta$ is given by the following equation:

[TABLE]

Then, $f_{k-1}$ is equal to the number of squarefree standard monomials of degree $k$. Recall that a squarefree monomial

[TABLE]

is standard if and only if $q_{A_{1}|\{1,2\}\cup B_{1}}\ldots q_{A_{\alpha}|\{1,2\}\cup B_{\alpha}}$ and $q_{\{1\}\cup A_{\alpha+1}^{\prime}|\{2\}\cup B_{\alpha+1}^{\prime}}\ldots q_{\{1\}\cup A_{k}^{\prime}|\{2\}\cup B_{k}^{\prime}}$ are standard. From Proposition 2.4, we have $f_{k-1}=\sum_{\alpha=0}^{k}B_{\alpha}C_{k-\alpha}$, where

[TABLE]

Since $B_{k}=0$ for any $k\geq n$, and $C_{k}=0$ for any $k\geq n-1$,

[TABLE]

Let $X=x-1$. From [14, Remark 2.4, Theorem 3.4], $h_{\Delta}(x)$ is a palindromic polynomial of degree $2n-5$. Hence,

[TABLE]

It then follows that

[TABLE]

Therefore, $h^{*}({\mathcal{P}},x)=h_{\Delta}(x)=(x+1)(A_{n-2}(x))^{2}$. ∎

4 Acknowledgment

The author is grateful to the anonymous referees for their careful reading and useful comments.