Let S be a finite alphabet. An injective word over S is a word over S such that each letter in S appears at most once in the word. We study Boolean cell complexes of injective words over S and their commutation classes. This generalizes work by Farmer and by Björner and Wachs on the complex of all injective words. Specifically, for an abstract simplicial complex ∆, we consider the Boolean cell ...

Trees with labelled leaves and with all other vertices of degree three play an important role in systematic biology and other areas of classification. A classical combinatorial result ensures that such trees can be uniquely reconstructed from the distances between the leaves (when the edges are given any strictly positive lengths). Moreover, a linear number of these pairwise distance values suf...

Hochschild lattices are specific intervals in the dexter meet-semilattices recently introduced by Chapoton. A natural geometric realization of these leads to some cell complexes Saneblidze, called polytopes. We obtain several geometrical properties lattices, namely we give cubic realizations, establish that EL-shellable, and show they constructible interval doubling. also prove combinatorial as...

let $r$ be a commutative ring with identity. let $g(r)$ denote the maximal graph associated to $r$, i.e., $g(r)$ is a graph with vertices as the elements of $r$, where two distinct vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $r$ containing both. let $gamma(r)$ denote the restriction of $g(r)$ to non-unit elements of $r$. in this paper we study the various graphi...

Let S be a finite alphabet. An injective word over S is a word over S such that each letter in S appears at most once in the word. We study Boolean cell complexes of injective words over S and their commutation classes. This generalizes work by Farmer and by Björner and Wachs on the complex of all injective words. Specifically, for an abstract simplicial complex ∆, we consider the Boolean cell ...

We show that an ’almost strong Lefschetz’ property holds for the barycentric subdivision of a shellable complex. From this we conclude that for the barycentric subdivision of a CohenMacaulay complex, the h-vector is unimodal, peaks in its middle degree (one of them if the dimension of the complex is even), and that its g-vector is an M -sequence. In particular, the (combinatorial) g-conjecture ...

All lattices are assumed to be finite. Björner [2] has shown that a dismantlable (see Rival, [5]) lattice L is Cohen-Macaulay (see [6] for definition) if and only if L is ranked and interval-connected. A lattice is planar if its Hasse diagram can be drawn in the plane with no edges crossing. Baker, Fishburn and Roberts have shown that planar lattices are dismantlable, see [1]. Lexicographically...

We consider a simplicial complex generaliztion of a result of Billera and Meyers that every nonshellable poset contains the smallest nonshellable poset as an induced subposet. We prove that every nonshellable 2-dimensional simplicial complex contains a nonshellable induced subcomplex with less than 8 vertices. We also establish CL-shellability of interval orders and as a consequence obtain a fo...

We investigate the conjectured sufficiency of a condition for h-vectors (1, h1, h2, . . . , hd, 0) of regular d-dimensional triangulations. (The condition is already shown to be necessary in [2]). We first prove that the condition is sufficient when h1 ≥ h2 ≥ · · · ≥ hd. We then derive some new shellings of squeezed spheres and use them to prove that the condition is sufficient when d = 3. Fina...

Let Cn = {compositions of n}, C = ∪Cn. We define a partial order making C into a ranked poset having 1 as its bottom element and Cn as its (n− 1)-st rank level. Let α = a1 + · · · + ak ∈ Cn. The interval [1, α] is shown to have the following properties: • The number of maximal chains in [1, α] equals the number of permutations of [n] with descent set {a1, a1 + a2, . . .}. • The interval [1, α] ...