Séminaire Lotharingien de Combinatoire, B45d (2001), 18 pp.

Helmut Krämer

Binary Moore-Penrose Inverses of Set Inclusion Incidence Matrices

Abstract. This note is a supplement to some recent work of R. B. Bapat on Moore-Penrose inverses of set inclusion matrices. Among other things Bapat constructs these inverses (in case of existence) for H(s,k) mod p, p an arbitrary prime, 0 <= s <= k <= v-s. Here we restrict ourselves to p=2. We give conditions for s,k which are easy to state and which ensure that the Moore-Penrose inverse of H(s,k) mod 2 equals its transpose. E.g., H(s,v-s) mod 2 has this property. Furthermore Ker H(s,v-s) mod 2 is nonzero if 0 < 2s < v <= 3s and then there is a decomposition

$\displaystyle {\rm Ker}\, H(s,v-s) \equiv \underset
{2 \mid \binom{v-s-j } { v-2s}} {\sum _
{0 \le j \le s-1}} {\rm Im}\,H(v-s,v-j)\ {\rm mod}\ 2.$

Also, refinements of this decomposition are given.

Received: December 14, 2000; Accepted: March 9, 2001.

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