ISRAEL JOURNAL OF MATHEMATICS 8"7 (1994), 77-87 A COMBINATORIAL DECOMPOSITION OF SIMPLICIAL COMPLEXES BY ART M. DUVAL Department of Mathematical Sciences University of Texas at El Paso El Paso, TX 79968-0514, USA e-mail: artduval@math.ep.utexas.edu ABSTRACT We find a decomposition of simplicial complexes that implies and sharp- ens the characterization (due to Bj6rner and Kalai) of the f-vector and Betti numbers of a simplicial complex. It generalizes a result of Stanley, who proved the a~yclic case, and settles a conjecture of Stanley and Kalai. 1. Introduction Let A be a finite (abstract) simplicial complex on vertex set V = (Xl,...,xn} (i.e. , A is a collection of subsets of V such that: V C_ A; and, if F C G and G E A, then F E A). Let the dimension of F E A be dirnF = IFI - 1, and the dimension of A be dirnA = max(dirnF: F E A}. Also let d = 1 + dirnA, so the largest face of A has d vertices. Let fi =/~(A) = #iF E A: dirnF = {}. In particular, f-1 = 1 for the empty set (unless A = 0), f0 counts the vertices ot A, and f~ -- 0 for i _> d. The f-vector of A is f(A) = (f0,..., fd-1). The same notion of fi(A) and the ]-vector will apply in this paper to every finite collection of sets. For a sirnplicial complex A, /~(A) = dirnKH~(A; K) will denote the ith (re- duced) Bettl number of A with respect to a fixed field of coefficients K, Received October 21, 1992 77