A Numerical Characterization of Gorenstein Complexes

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 = #(V) and d = dim .1 + 1. Write /; = /;(.1) for the number of i-faces of .1. Thus, in particular, fo = v. The vector f = f(L1) = (fo,/t, ... ,fd-l) is called the f-vector of .1. In terms of the f-vector, letting f-l = 1, define