Counting of Moore Families for n=7

, , and . page 72--87. Springer Berlin Heidelberg, Berlin, Heidelberg, (2010)


Given a set U n þinspace=þinspace\0,1,...,nþinspace−þinspace1\, a collection \$\backslashmathcal\M\\$ of subsets of U n that is closed under intersection and contains U n is known as a Moore family. The set of Moore families for a given n, denoted by M n , increases very quickly with n, thus |M 3| is 61 and |M 4| is 2480. In 1 the authors determined the number for nþinspace=þinspace6 and stated a 24h- computation-time. Thus, the number for nþinspace=þinspace7 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 nþinspace=þinspace7 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.

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