## Counting of Moore families for $$n=7$$.(English)Zbl 1274.05013

Kwuida, Léonard (ed.) et al., Formal concept analysis. 8th international conference, ICFCA 2010, Agadir, Morocco, March 15–18, 2010. Proceedings. Berlin: Springer (ISBN 978-3-642-11927-9/pbk). Lecture Notes in Computer Science 5986. Lecture Notes in Artificial Intelligence, 72-87 (2010).
Summary: Given a set $$U_n = \{ 0, 1, \dots, n-1 \}$$, a collection $$\mathcal{M}$$ 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 [M. Habib and L. Nourine, Discrete Math. 294, No. 3, 291–296 (2005; Zbl 1083.06003)], 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.
For the entire collection see [Zbl 1184.68008].

### MSC:

 05A15 Exact enumeration problems, generating functions 06A15 Galois correspondences, closure operators (in relation to ordered sets)

Zbl 1083.06003
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