Counting of moore families for n=7

Authors:
Pierre Colomb;Alexis Irlande;Olivier Raynaud
Affiliations:
Université Blaise Pascal, Aubiére, France;Universidad Nacional de Colombia, Bogota, Colombia;Université Blaise Pascal, Aubiére, France
Venue:
ICFCA'10 Proceedings of the 8th international conference on Formal Concept Analysis
Year:
2010

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Abstract

Given a set Un={0,1,...,n−1}, a collection $\mathcal{M}$ 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.