%I #22 Aug 05 2019 07:36:32

%S 2,4,10,38,368,29328,216591692,5592326399531792

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

%C 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 Also the number of non-isomorphic sets of subsets of {1..n} that are closed under union. - _Gus Wiseman_, Aug 04 2019

%D D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 79.

%H 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 D. E. Knuth, <a href="http://www-cs-faculty.stanford.edu/~knuth/programs.html">HORN-COUNT</a>

%F a(n) = 2 * A193674(n).

%e From _Gus Wiseman_, Aug 04 2019: (Start)

%e Non-isomorphic representatives of the a(0) = 2 through a(2) = 10 sets of sets:

%e {} {} {}

%e {{}} {{}} {{}}

%e {{1}} {{1}}

%e {{},{1}} {{1,2}}

%e {{},{1}}

%e {{},{1,2}}

%e {{2},{1,2}}

%e {{},{2},{1,2}}

%e {{1},{2},{1,2}}

%e {{},{1},{2},{1,2}}

%e (End)

%Y The covering case is A326907.

%Y The case without {} is A193674.

%Y The labeled version is A102897.

%Y The same with intersection instead of union is also A193675.

%Y The case closed under both union and intersection also is A326908.

%Y Cf. A102894, A102895, A102896, A102897, A108798, A108800, A326867, A326875, A326904.

%K nonn,hard,nice,more

%O 0,1

%A _Don Knuth_, Jul 01 2005

%E a(6) received from _Don Knuth_, Aug 17 2005

%E a(6) corrected by Pierre Colomb, Aug 02 2011

%E a(7) = 2*A193674(7) from _Hugo Pfoertner_, Jun 18 2018