Takayuki Hibi

What can be said about pure O -sequences?

Abstract Let f(Δ) = (f0, f1, …, fd−1) be the f-vector of a Cohen-Macaulay complex Δ. Bjorner proved that (∗) fi⩽f(d−2)−i for any 0⩽i d 2 ] and (∗∗) f 0 ⩽f 1 ⩽ … ⩽f [ (d−1) 2 ] . Recently, Stanley generalized Bjorners inequalities (∗) and (∗∗) for pure simplicial complexes. In this paper we consider O-sequence analogue of the inequalities (∗) and (∗∗). Let (h0, h1, …, hs), hs≠0, is a pure O-sequence. We shall prove that hi⩽hs−i for any 0⩽i⩽[ s 2 ] and h 0 ⩽h 1 ⩽ … ⩽h [ (s+1) 2 ] .

Level rings and algebras with straightening laws

Abstract Stanley [13] defined the concept of “level rings,” intermediate between Cohen-Macaulay and Gorenstein, which is of interest from viewpoints of both commutative algebra and combinatorics. Via the theory of canonical modules, the study of level rings turns out to be one of the most important foundations of the theory of algebras with straightening laws. The purpose of this paper is to state some basic results on level rings which are either Stanley-Reisner rings or algebras with straightening laws.

Quotient algebras of Stanley-Reisner rings and local cohomology

We study certain subcomplexes Δ′ of an arbitrary simplicial complex Δ such that Hmi(k[Δ])∼-Hmi(k[Δ′]) for any 0⩽i<dim(k[Δ′]). Here, Hmi(k[Δ]) is the ith local cohomology module of the Stanley-Reisner ring k[Δ] of Δ over a field k. Our technique is an elegant approach to one of the most generalized versions of the rank selection theorems of J. Munkres (1984, Michigan Math. J.31, 113–128, Theorem 6.4) and R. Stanley (1979, Trans. Amer. Math. Soc.249, 139–157, Theorem 4.3).

A recurrence for linear extensions

The number e(P) of linear extensions of a finite poset P is expressed in terms of e(Q) for certain smaller posets Q. The proof is based on M. Schützengergers concept of promotions of linear extensions.

Flawless 0-sequences and Hilbert functions of Cohen-Macaulay integral domains

Abstract Let A = A 0 ⊗ A 1 ⊗⋯ be a commutative graded ring such that (i) A 0 = k a field, (ii) A = k [ A 1 ] and (iii) dim k A 1 ∞ n = 0 (dim k A n ) λ n is of the form (h 0 + h 1 λ + ⋯ + h s λ s ) (1 − λ) dim A with each h i ϵ Z . We are interested in the sequence ( h 0 , h 1 ,…, h s ), called the h -vector of A , when A is a Cohen–Macaulay integral domain. In this paper, after summarizing fundamental results (Section 1), we study h -vectors of certain Gorenstein domains (Section 2) and find some examples of h -vectors arising from integrally closed level domains (Sections 3 and 4).

Hilbert functions of Cohen— Macaulay integral domains and chain conditions of finite partially ordered sets☆

Let A = A0 ⊕ A1 ⊕ ⋯ be a commutative graded ring such that (i) A0 = k a field. (ii) A = k[A1] and (iii) dimkA1 < ∞. It is a fundamental fact that the formal power series ∑∗n=0(dimλ An)λn is of the form (h0 + h1λ + ⋯ + h8λx)/(1-λ) dim A, where each h, is an integer. We are interested in the sequence (h0,h1,…,hx) called tL. h-vector of A, when A is a Cohen-Macaulay integral domain. In this paper, first, in Section 1, we summarize the basic information on the h-vector of A which can be obtained by investigating the behavior of the canonical module KA of A. Secondly, in Section 2, we apply the abstract theory in Section 1 to the combinatorial problem of finding a characterization of the so-called w-vectors of finite partially ordered sets and obtain linear and nonlinear inequalities for the w-vector of a finite partially ordered set which satisfies a certain chain condition.

Divisor class groups of affine semigroup rings associated with distributive lattices

Abstract We compute divisor class groups of certain normal affine semigroup rings associated with distributive lattices by means of exact sequences on toric varieties.

Linear and nonlinear inequalities concerning a certain combinatorial sequence which arises from counting the number of chains of finite distributive lattice

Abstract Let f i = f i (L), 0≤i≤n , be the number of chains (totally ordered sets) of length i contained in a finite distributive lattice L of rank n , and set f −1 = 1. We study the sequence w 0 , w 1 ,…, w s ( w s ≠0) defined by the formula ∑ n + 1 i = 0 f i − 1 ( x − 1) n + 1 − i = ∑ s i = 0 w i x n + 1 − i .

Plane Graphs and Cohen-Macaulay Posets

For which plane graphs G allowing loops and multiple edges is the face poset P(G) of G Cohen-Macaulay (resp. level, Gorenstein)? The purpose of this paper is to prove the following results. (C-M) The poset P(G) is Cohen-Macaulay iff G is connected. (Lev) The poset P(G) is level iff (i) G is connected and (ii) either G has no loop and no cut-vertex or G-{loops} is a tree. Also, (Gor) The poset P(G) is Gorenstein iff (i) G is connected and (ii) either G has no loop and no cut-vertex or G consists of (a) one vertex and one loop or (b) two vertices and one edge.

Classification of integral trees

A finite poset Q is called integral over a field k if there exists an ASL (algebra with straightening laws) domain on Q∪{−∞} over k. We classify all ‘trees’ (rank one connected posets without cycles) which are integral over an infinite field.