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A193674 Number of nonisomorphic systems enumerated by A102896; that is, the number of inequivalent closure operators (or Moore families).

%I A193674 #46 Mar 20 2020 15:07:31
%S A193674 1,2,5,19,184,14664,108295846,2796163199765896
%N A193674 Number of nonisomorphic systems enumerated by A102896; that is, the number of inequivalent closure operators (or Moore families).
%C A193674 Also the number of unlabeled n-vertex set-systems (A003180) closed under union. - _Gus Wiseman_, Aug 01 2019
%D A193674 D. E. Knuth, The Art of Computer Programming, Vol. 4, Section 7.1.1
%H A193674 Daniel Borchmann, Bernhard Ganter, <a href="https://doi.org/10.1007/978-3-642-01815-2_2">Concept Lattice Orbifolds - First Steps</a>, Proceedings of the 7th International Conference on Formal Concept Analysis (ICFCA 2009), 22-37 (Reference points to A108799).
%H A193674 G. Brinkmann and R. Deklerck, <a href="https://arxiv.org/abs/1701.03751">Generation of Union-Closed Sets and Moore Families</a>, arXiv:1701.03751 [math.CO], 2017.
%H A193674 G. Brinkmann and R. Deklerck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Brinkmann/brink6.html"> Generation of Union-Closed Sets and Moore Families</a>, Journal of Integer Sequences, Vol.21 (2018), Article 18.1.7.
%H A193674 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).
%F A193674 a(n) = A193675(n)/2.
%e A193674 From _Gus Wiseman_, Aug 01 2019: (Start)
%e A193674 Non-isomorphic representatives of the a(0) = 1 through a(3) = 19 set-systems closed under union:
%e A193674   {}  {}     {}               {}
%e A193674       {{1}}  {{1}}            {{1}}
%e A193674              {{1,2}}          {{1,2}}
%e A193674              {{2},{1,2}}      {{1,2,3}}
%e A193674              {{1},{2},{1,2}}  {{2},{1,2}}
%e A193674                               {{3},{1,2,3}}
%e A193674                               {{1},{2},{1,2}}
%e A193674                               {{2,3},{1,2,3}}
%e A193674                               {{1},{2,3},{1,2,3}}
%e A193674                               {{3},{2,3},{1,2,3}}
%e A193674                               {{1,3},{2,3},{1,2,3}}
%e A193674                               {{2},{3},{2,3},{1,2,3}}
%e A193674                               {{2},{1,3},{2,3},{1,2,3}}
%e A193674                               {{3},{1,3},{2,3},{1,2,3}}
%e A193674                               {{1,2},{1,3},{2,3},{1,2,3}}
%e A193674                               {{2},{3},{1,3},{2,3},{1,2,3}}
%e A193674                               {{3},{1,2},{1,3},{2,3},{1,2,3}}
%e A193674                               {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
%e A193674                               {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
%e A193674 (End)
%Y A193674 Cf. A102894, A102895, A102897.
%Y A193674 The labeled case is A102896.
%Y A193674 The covering case is A108798.
%Y A193674 The same for intersection instead of union is A108800.
%Y A193674 The case with empty edges allowed is A193675.
%Y A193674 Cf. A000612, A001930, A003180, A306445, A326875, A326883.
%K A193674 nonn,hard,more
%O A193674 0,2
%A A193674 _Don Knuth_, Jul 01 2005
%E A193674 a(6) received Aug 17 2005
%E A193674 a(6) corrected by Pierre Colomb, Aug 02 2011
%E A193674 a(7) from _Gunnar Brinkmann_, Feb 07 2018