Given a simplicial hyperplane arrangement H and a subspace arrangement A embedded in H, we define a simplicial complex ∆A,H as the subdivision of the link of A induced by H. In particular, this generalizes Steingŕımsson’s coloring complex of a graph. We do the following: (1) When A is a hyperplane arrangement, ∆A,H is shown to be shellable. As a special case, we answer affirmatively a question ...

We introduce pretty clean modules, extending the notion of clean modules by Dress, and show that pretty clean modules are sequentially CohenMacaulay. We also extend a theorem of Dress on shellable simplicial complexes to multicomplexes.

Each positive rational number x > 0 can be written uniquely as x = a/(b − a) for coprime positive integers 0 < a < b. We will identify x with the pair (a, b). In this extended abstract we use rational Dyck paths to define for each positive rational x > 0 a simplicial complex Ass(x) = Ass(a, b) called the rational associahedron. It is a pure simplicial complex of dimension a− 2, and its maximal ...

The set of all permutations, ordered by pattern containment, is a poset. We present an order isomorphism from the poset of permutations with a fixed number of descents to a certain poset of words with subword order. We use this bijection to show that intervals of permutations with a fixed number of descents are shellable, and we present a formula for the Möbius function of these intervals. We p...

To every directed graph G one can associate a complex (G) consisting of directed subforests. This construction, suggested to us by R. Stanley, is especially important in the case of a complete double directed graph Gn, where it leads to studying some interesting representations of the symmetric group and corresponds (via Stanley-Reisner correspondence) to an interesting quotient ring. Our main ...

Let Φ be a finite root system of rank n and let m be a positive integer. It is proved that the generalized cluster complex ∆m(Φ), introduced by S. Fomin and N. Reading, is (m + 1)-Cohen-Macaulay, in the sense of Baclawski. This statement was conjectured by V. Reiner. More precisely, it is proved that the simplicial complex obtained from ∆m(Φ) by removing any subset of its vertex set of cardinal...

Let I ⊂ K[x1, . . . , xn] be a zero-dimensional monomial ideal, and ∆(I) be the simplicial complex whose Stanley–Reisner ideal is the polarization of I. It follows from a result of Soleyman Jahan that ∆(I) is shellable. We give a new short proof of this fact by providing an explicit shelling. Moreover, we show that ∆(I) is even vertex decomposable. The ideal L(I), which is defined to be the Sta...