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A193675 Number of nonisomorphic systems enumerated by A102897; that is, the number of inequivalent Horn functions, under permutation of variables.

%I A193675 #22 Aug 05 2019 07:36:32
%S A193675 2,4,10,38,368,29328,216591692,5592326399531792
%N A193675 Number of nonisomorphic systems enumerated by A102897; that is, the number of inequivalent Horn functions, under permutation of variables.
%C A193675 When speaking of inequivalent Boolean functions, three groups of symmetries are typically considered: Complementations only, the Abelian group (2,...,2) of 2^n elements; permutations only, the symmetric group of n! elements; or both complementations and permutations, the octahedral group of 2^n n! elements. In this case only symmetry with respect to the symmetric group is appropriate because complementation affects the property of being a Horn function.
%C A193675 Also the number of non-isomorphic sets of subsets of {1..n} that are closed under union. - _Gus Wiseman_, Aug 04 2019
%D A193675 D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 79.
%H A193675 P. Colomb, A. Irlande and O. Raynaud, <a href="http://pierre.colomb.me/data/paper/icfca2010.pdf">Counting of Moore Families for n=7</a>, International Conference on Formal Concept Analysis (2010).
%H A193675 D. E. Knuth, <a href="http://www-cs-faculty.stanford.edu/~knuth/programs.html">HORN-COUNT</a>
%F A193675 a(n) = 2 * A193674(n).
%e A193675 From _Gus Wiseman_, Aug 04 2019: (Start)
%e A193675 Non-isomorphic representatives of the a(0) = 2 through a(2) = 10 sets of sets:
%e A193675   {}    {}        {}
%e A193675   {{}}  {{}}      {{}}
%e A193675         {{1}}     {{1}}
%e A193675         {{},{1}}  {{1,2}}
%e A193675                   {{},{1}}
%e A193675                   {{},{1,2}}
%e A193675                   {{2},{1,2}}
%e A193675                   {{},{2},{1,2}}
%e A193675                   {{1},{2},{1,2}}
%e A193675                   {{},{1},{2},{1,2}}
%e A193675 (End)
%Y A193675 The covering case is A326907.
%Y A193675 The case without {} is A193674.
%Y A193675 The labeled version is A102897.
%Y A193675 The same with intersection instead of union is also A193675.
%Y A193675 The case closed under both union and intersection also is A326908.
%Y A193675 Cf. A102894, A102895, A102896, A102897, A108798, A108800, A326867, A326875, A326904.
%K A193675 nonn,hard,nice,more
%O A193675 0,1
%A A193675 _Don Knuth_, Jul 01 2005
%E A193675 a(6) received from _Don Knuth_, Aug 17 2005
%E A193675 a(6) corrected by Pierre Colomb, Aug 02 2011
%E A193675 a(7) = 2*A193674(7) from _Hugo Pfoertner_, Jun 18 2018