Let V be a finite set, called the vertex set, and let .1 be a simplicial complex on V. Thus .1 is a collection of subsets of V such that (i) {x} EL1 for any x E V, and (ii) a E .1, 'r c a imply 'r E L1. An element a of .1 is called an i-face if #( a) = i + 1. Here, #( a) is the cardinality of a as a set. The positive integer dim .1: = max{ #( a) 1; a E L1} is called the dimension of .1. Let v =...

All dissections of a convex (mn + 2)-gons into (m + 2)-gons are facets of a simplicial complex. This complex is introduced by S. Fomin and A.V. Zelevinsky in [7]. We reprove the result of E. Tzanaki about shellability of such complex by finding a concrete shelling order. Also, we use this shelling order to find a combinatorial interpretation of h-vector and to describe the generating facets of ...

We prove that it is NP-complete to decide whether a given (3-dimensional) simplicial complex is collapsible. This work extends a result of Malgouyres and Francés showing that it is NP-complete to decide whether a given simplicial complex collapses to a 1-complex.

For any positive integer n, let [n] denote the set {1, . . . , n}, and letMn be the set of all matroids on [n]. Throughout this paper all matroids will have ground set [n], and we shall frequently omit the symbol n from our notation. Define a partial ordering on Mn by M ′ ≤ M if M ′ is a quotient of M . Let Ωn be the simplicial complex of chains in Mn ; every simplex s ∈ Ωn can be written as s ...

Given a simplicial group G, there are two known classifying simplicial set constructions, the Kan classifying simplicial set WG and Diag NG, where N denotes the dimensionwise nerve. They are known to be weakly homotopy equivalent. We will show that WG is a strong simplicial deformation retract of Diag NG. In particular, WG and Diag NG are simplicially homotopy equivalent.

Nested set complexes appear as the combinatorial core of De ConciniProcesi arrangement models. We show that nested set complexes are homotopy equivalent to the order complexes of the underlying meet-semilattices without their minimal elements. For atomic semilattices, we consider the realization of nested set complexes by simplicial fans proposed in [FY], and we strengthen our previous result s...

A very interesting abstract simplicial complex T (k) n has faces in bijection with the trees with at most n interior vertices, all of which have degrees at least k+2 and are congruent to 2 mod k, and whose leaves are labelled by the distinct integers in [0, 1, ..., m], where m+1 :=nk+2 is the number of leaves (n 0, k 1). Thus the facets of T n correspond to the leaf-labelled trees with n interi...

Pipe dreams represent permutations pictorially as a series of crossing pipes. Recent applications of pipe dreams include the calculation of Schubert polynomials, fillings of moon polyominoes, and in the combinatorics of antidiagonal simplicial complexes. These applications associate pipe dreams to words of elementary symmetric transpositions via a canonical mapping. However, this canonical mapp...

W. Fulton–R. MacPherson [15] found a Sullivan dg-algebra model for the space of n-configurations of a smooth compact complex algebraic variety X . I. Kř́ıž [16] gave a simpler model, En(H), depending only on the cohomology ring, H := H X . We construct an even simpler and smaller model, Jn(H). We then define another new dg-algebra, En( o H), and use Jn(H) to prove that En( o H) is a model of the...