Moore–Penrose inverse of set inclusion matrices

Abstract Given integers s,k and v, let W sk be the v s × v k 0–1 matrix, the rows and the columns of which are indexed by the s -subsets and the k -subsets of a v -set respectively, and where the entry in row S and column U is 1 if S⊂U and 0 otherwise. A formula for the Moore–Penrose inverse of W sk over the reals is obtained. A necessary and sufficient condition for W sk to admit a Moore–Penrose inverse over the set of integers modulo a prime p is given, together with a formula for the Moore–Penrose inverse when it exists.