Counting of Moore families for n=7

Abstract

Given a set Un = {0, 1, ..., n-1}, a collection M of subsets of Un that is closed under intersection and contains Un is known as a Moore family. The set of Moore families for a given n, denoted by Mn, increases very quickly with n, thus |M3| is 61 and |M4| is 2480. In [1] the authors determined the number for n = 6 and stated a 24h- computation-time. Thus, the number for n = 7 can be considered as an extremely difficult technical challenge. In this paper, we introduce a counting strategy for determining the number of Moore families for n = 7 and we give the exact value : 14 087 648 235 707 352 472. Our calculation is particularly based on the enumeration of Moore families up to an isomorphism for n ranging from 1 to 6. © Springer-Verlag Berlin Heidelberg 2010.