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Akiyoshi Tsuchiya

Publications and source records attributed to Akiyoshi Tsuchiya.

At least 55 records · Page 3Linked to original sources
arXiv2018

Levelness of Order Polytopes

Since their introduction by Stanley~\cite{StanleyOrderPoly} order polytopes have been intriguing mathematicians as their geometry can be used to examine (algebraic) properties of finite posets. In this paper, we follow this route to examine the levelness property of order polytopes. The levelness property was also introduced by Stanley~\cite{Stanley-CM-complexes} and it generalizes the Gorenstein property. This property has been recently characterized by Miyazaki~\cite{Miyazaki} for the case of order polytopes. We provide an alternative characterization using weighted digraphs. Using this characterization, we give a new infinite family of level posets and show that determining levelness is in $\operatorname{co-NP}$. This family can be used to create infinitely many examples illustrating that the levelness property can not be characterized by the $h^{\ast}$-vector. We then turn to the more general family of alcoved polytopes. We give a characterization for levelness of alcoved polytopes using the Minkowski sum. Then we study several cases when the product of two polytopes is level. In particular, we provide an example where the product of two level polytopes is not level.

math.CO↗
arXiv2018

Reflexive polytopes arising from perfect graphs

Reflexive polytopes form one of the distinguished classes of lattice polytopes. Especially reflexive polytopes which possess the integer decomposition property are of interest. In the present paper, by virtue of the algebraic technique on Grönbner bases, a new class of reflexive polytopes which possess the integer decomposition property and which arise from perfect graphs will be presented. Furthermore, the Ehrhart $δ$-polynomials of these polytopes will be studied.

math.CO↗
arXiv2018

Reflexive polytopes arising from partially ordered sets and perfect graphs

Reflexive polytopes which have the integer decomposition property are of interest. Recently, some large classes of reflexive polytopes with integer decomposition property coming from the order polytopes and the chain polytopes of finite partially ordered sets are known. In the present paper, we will generalize this result. In fact, by virtue of the algebraic technique on Gröbner bases, new classes of reflexive polytopes with the integer decomposition property coming from the order polytopes of finite partially ordered sets and the stable set polytopes of perfect graphs will be introduced. Furthermore, the result will give a polyhedral characterization of perfect graphs. Finally, we will investigate the Ehrhart $δ$-polynomials of these reflexive polytopes.

math.CO↗
arXiv2018

Volume, facets and dual polytopes of twinned chain polytopes

Let $(P,\leq_P)$ and $(Q,\leq_Q)$ be finite partially ordered sets with $|P|=|Q|=d$, and $\mathcal{C}(P) \subset \mathbb{R}^d$ and $\mathcal{C}(Q) \subset \mathbb{R}^d$ their chain polytopes. The twinned chain polytope of $P$ and $Q$ is the lattice polytope $Γ(\mathcal{C}(P),\mathcal{C}(Q)) \subset \mathbb{R}^d$ which is the convex hull of $\mathcal{C}(P) \cup (-\mathcal{C}(Q))$. It is known that twinned chain polytopes are Gorenstein Fano polytopes with the integer decomposition property. In the present paper, we study combinatorial properties of twinned chain polytopes. First, we will give the formula of the volume of twinned chain polytopes in terms of the underlying partially ordered sets. Second, we will identify the facet-supporting hyperplanes of twinned chain polytopes in terms of the underlying partially ordered sets. Finally, we will provide the vertex representations of the dual polytopes of twinned chain polytopes.

math.CO↗
arXiv2017

Laplacian Simplices Associated to Digraphs

We associate to a finite digraph $D$ a lattice polytope $P_D$ whose vertices are the rows of the Laplacian matrix of $D$. This generalizes a construction introduced by Braun and the third author. As a consequence of the Matrix-Tree Theorem, we show that the normalized volume of $P_D$ equals the complexity of $D$, and $P_D$ contains the origin in its relative interior if and only if $D$ is strongly connected. Interesting connections with other families of simplices are established and then used to describe reflexivity, $h^*$-polynomial, and integer decomposition property of $P_D$ in these cases. We extend Braun and Meyer's study of cycles by considering cycle digraphs. In this setting we characterize reflexivity and show there are only four non-trivial reflexive Laplacian simplices having the integer decomposition property.

math.CO↗
arXiv2017

Gorenstein simplices and the associated finite abelian groups

It is known that a lattice simplex of dimension $d$ corresponds a finite abelian subgroup of $(\mathbb{R}/\mathbb{Z})^{d+1}$. Conversely, given a finite abelian subgroup of $(\mathbb{R}/\mathbb{Z})^{d+1}$ such that the sum of all entries of each element is an integer, we can obtain a lattice simplex of dimension $d$. In this paper, we discuss a characterization of Gorenstein simplices in terms of the associated finite abelian groups. In particular, we present complete characterizations of Gorenstein simplices whose normalized volume equals $p,p^2$ and $pq$, where $p$ and $q$ are prime numbers with $p \neq q$. Moreover, we compute the volume of the dual simplices of Gorenstein simplices.

math.CO↗
arXiv2017

Self dual reflexive simplices with Eulerian polynomials

A lattice polytope $\mathcal{P}$ is called reflexive if its dual $\mathcal{P}^\vee$ is a lattice polytope. The property that $\mathcal{P}$ is unimodularly equivalent to $\mathcal{P}^\vee$ does not hold in general, and in fact there are few examples of such polytopes. In this note, we introduce a new reflexive simplex $Q_n$ which has this property. Additionally, we show that $δ$-polynomalial of $Q_n$ is the Eulerian polynomial and show the existence of a regular, flag, unimodular triangulation.

math.CO↗
arXiv2017

Facets and volume of Gorenstein Fano polytopes

It is known that every integral convex polytope is unimodularly equivalent to a face of some Gorenstein Fano polytope. It is then reasonable to ask whether every normal polytope is unimodularly equivalent to a face of some normal Gorenstein Fano polytope. In the present paper, it is shown that, by giving new classes of normal Gorenstein Fano polytopes, each order polytope as well as each chain polytope of dimension $d$ is unimodularly equivalent to a facet of some normal Gorenstein Fano polytopes of dimension $d + 1$. Furthermore, investigation on combinatorial properties, especially, Ehrhart polynomials and volume of these new polytopes will be achieved. Finally, some curious examples of Gorenstein Fano polytopes will be discovered.

math.CO↗
arXiv2017

Flat $δ$-vectors and their Ehrhart polynomials

We call the $δ$-vector of an integral convex polytope of dimension $d$ flat if the $δ$-vector is of the form $(1,0,\ldots,0,a,\ldots,a,0,\ldots,0)$, where $a \geq 1$. In this paper, we give the complete characterization of possible flat $δ$-vectors. Moreover, for an integral convex polytope $\mathcal{P} \subset \mathbb{R}^N$ of dimension $d$, we let $i(\mathcal{P},n)=|n\mathcal{P} \cap \mathbb{Z}^N|$ and $\ i^*(\mathcal{P},n)=|n(\mathcal{P} \setminus \partial \mathcal{P}) \cap \mathbb{Z}^N|.$ By this characterization, we show that for any $d \geq 1$ and for any $k,\ell \geq 0$ with $k+\ell \leq d-1$, there exist integral convex polytopes $\mathcal{P}$ and $\mathcal{Q}$ of dimension $d$ such that (i) For $t=1,\ldots,k$, we have $i(\mathcal{P},t)=i(\mathcal{Q},t),$ (ii) For $t=1,\ldots,\ell$, we have $i^*(\mathcal{P},t)=i^*(\mathcal{Q},t)$ and (iii) $i(\mathcal{P},k+1) \neq i(\mathcal{Q},k+1)$ and $i^*(\mathcal{P},\ell+1)\neq i^*(\mathcal{Q},\ell+1).$

math.CO↗
arXiv2016

Nonincreasing depth functions of monomial ideals

Given a nonincreasing function $f : \mathbb{Z}_{\geq 0} \setminus \{ 0 \} \to \mathbb{Z}_{\geq 0}$ such that (i) $f(k) - f(k+1) \leq 1$ for all $k \geq 1$ and (ii) if $a = f(1)$ and $b = \lim_{k \to \infty} f(k)$, then $|f^{-1}(a)| \leq |f^{-1}(a-1)| \leq \cdots \leq |f^{-1}(b+1)|$, a system of generators of a monomial ideal $I \subset K[x_1, \ldots, x_n]$ for which ${\rm depth} S/I^k = f(k)$ for all $k \geq 1$ is explicitly described. Furthermore, we give a characterization of triplets of integers $(n,d,r)$ with $n > 0$, $d \geq 0$ and $r > 0$ with the properties that there exists a monomial ideal $I \subset S = K[x_1, \ldots, x_n]$ for which $\lim_{k \to \infty} {\rm depth} S/I^k = d$ and ${\rm dstab}(I) = r$, where ${\rm dstab}(I)$ is the smallest integer $k_0 \geq 1$ with ${\rm depth} S/I^{k_0} = {\rm depth} S/I^{k_0+1} = {\rm depth} S/I^{k_0+2} = \cdots$.

math.AC↗
arXiv2016

Gorenstein properties and integer decomposition properties of lecture hall polytopes

Though much is known about ${\bf s}$-lecture hall polytopes, there are still many unanswered questions. In this paper, we show that ${\bf s}$-lecture hall polytopes satisfy the integer decomposition property (IDP) in the case of monotonic ${\bf s}$-sequences. Given restrictions on a monotonic ${\bf s}$-sequence, we discuss necessary and sufficient conditions for the Fano, reflexive and Gorenstein properties. Additionally, we give a construction for producing Gorenstein/IDP lecture hall polytopes.

math.CO↗
arXiv2016

Order-Chain Polytopes

Given two families $X$ and $Y$ of integral polytopes with nice combinatorial and algebraic properties, a natural way to generate new class of polytopes is to take the intersection $\mathcal{P}=\mathcal{P}_1\cap\mathcal{P}_2$, where $\mathcal{P}_1\in X$, $\mathcal{P}_2\in Y$. Two basic questions then arise: 1) when $\mathcal{P}$ is integral and 2) whether $\mathcal{P}$ inherits the "old type" from $\mathcal{P}_1, \mathcal{P}_2$ or has a "new type", that is, whether $\mathcal{P}$ is unimodularly equivalent to some polytope in $X\cup Y$ or not. In this paper, we focus on the families of order polytopes and chain polytopes and create a new class of polytopes following the above framework, which are named order-chain polytopes. In the study on their volumes, we discover a natural relation with Ehrenborg and Mahajan's results on maximizing descent statistics.

math.CO↗
arXiv2016

The $δ$-vectors of reflexive polytopes and of the dual polytopes

Let $δ(\mathcal{P})$ be the $δ$-vector of a reflexive polytope $\mathcal{P} \subset \mathbb{R}^d$ of dimension $d$ and $δ(\mathcal{P} ^\vee)$ the $δ$-vector of the dual polytope $\mathcal{P}^\vee \subset \mathbb{R}^d$. In general, $δ(\mathcal{P})=δ(\mathcal{P}^\vee)$ does not hold. In this paper, we give a higher-dimensional construction of reflexive polytope whose $δ$-vector equals the $δ$-vector of the dual polytope. In particular, we consider the case that the reflexive polytope and the dual polytope are unimodularly equivalent.

math.CO↗
arXiv2016

Ehrhart polynomials with negative coefficients

It is shown that, for each $d \geq 4$, there exists an integral convex polytope $\mathcal{P}$ of dimension $d$ such that each of the coefficients of $n, n^{2}, \ldots, n^{d-2}$ of its Ehrhart polynomial $i(\mathcal{P},n)$ is negative. Moreover, it is also shown that for each $d \geq 3$ and $1 \leq k \leq d-2$, there exists an integral convex polytope $\mathcal{P}$ of dimension $d$ such that the coefficient of $n^k$ of the Ehrhart polynomial $i(\mathcal{P},n)$ of $\mathcal{P}$ is negative and all its remaining coefficients are positive. Finally, we consider all the possible sign patterns of the coefficients of the Ehrhart polynomials of low dimensional integral convex polytopes.

math.CO↗
arXiv2015

Quadratic Gröbner bases arising from partially ordered sets

The order polytope $\mathcal{O}(P)$ and the chain polytope $\mathcal{C}(P)$ associated to a partially ordered set $P$ are studied. In this paper, we introduce the convex polytope $Γ(\mathcal{O}(P), -\mathcal{C}(Q))$ which is the convex hull of $\mathcal{O}(P) \cup (-\mathcal{C}(Q))$, where both $P$ and $Q$ are partially ordered sets with $|P|=|Q|=d$. It will be shown that $Γ(\mathcal{O}(P), -\mathcal{C}(Q))$ is a normal and Gorenstein Fano polytope by using the theory of reverse lexicographic squarefree initial ideals of toric ideals.

math.AC↗
arXiv2015

Dominating induced matchings of finite graphs and regularity of edge ideals

The regularity of an edge ideal of a finite simple graph $G$ is at least the induced matching number of $G$ and is at most the minimum matching number of $G$. If $G$ possesses a dominating inuduced matching, i.e., an induced matching which forms a maximal matching, then the induced matching number of $G$ is equal to the minimum matching number of $G$. In the present paper, from viewpoints of both combinatorics and commutative algebra, finite simple graphs with dominating induced matchings will be mainly studied.

math.CO↗
arXiv2015

Gorenstein Fano polytopes arising from order polytopes and chain polytopes

Richard Stanley introduced the order polytope $\mathcal{O}(P)$ and the chain polytope $\mathcal{C}(P)$ arising from a finite partially ordered set $P$, and showed that the Ehrhart polynomial of $\mathcal{O}(P)$ is equal to that of $\mathcal{C}(P)$. In addition, the unimodular equivalence problem of $\mathcal{O}(P)$ and $\mathcal{C}(P)$ was studied by the first author and Nan Li. In the present paper, three integral convex polytopes $Γ(\mathcal{O}(P), -\mathcal{O}(Q))$, $Γ(\mathcal{O}(P), -\mathcal{C}(Q))$ and $Γ(\mathcal{C}(P), -\mathcal{C}(Q))$, where $P$ and $Q$ are partially ordered sets with $| P | = | Q |$, will be studied. First, it will be shown that the Ehrhart polynomial of $Γ(\mathcal{O}(P), -\mathcal{C}(Q))$ coincides with that of $Γ(\mathcal{C}(P), -\mathcal{C}(Q))$. Furthermore, when $P$ and $Q$ possess a common linear extension, it will be proved that these three convex polytopes have the same Ehrhart polynomial. Second, the problem of characterizing partially ordered sets $P$ and $Q$ for which $Γ(\mathcal{O}(P), -\mathcal{O}(Q))$ or $Γ(\mathcal{O}(P), -\mathcal{C}(Q))$ or $Γ(\mathcal{C}(P), -\mathcal{C}(Q))$ is a smooth Fano polytope will be solved. Finally, when these three polytopes are smooth Fano polytopes, the unimodular equivalence problem of these three polytopes will be discussed.

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