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

1, 2, 5, 19, 184, 14664, 108295846, 2796163199765896

Offset: 0

Author

Don Knuth, Jul 01 2005

Keywords

• nonn
• hard
• more

Comments

Also the number of unlabeled n-vertex set-systems (A003180) closed under union. - Gus Wiseman, Aug 01 2019

Examples

			From _Gus Wiseman_, Aug 01 2019: (Start)
Non-isomorphic representatives of the a(0) = 1 through a(3) = 19 set-systems closed under union:
  {}  {}     {}               {}
      {{1}}  {{1}}            {{1}}
             {{1,2}}          {{1,2}}
             {{2},{1,2}}      {{1,2,3}}
             {{1},{2},{1,2}}  {{2},{1,2}}
                              {{3},{1,2,3}}
                              {{1},{2},{1,2}}
                              {{2,3},{1,2,3}}
                              {{1},{2,3},{1,2,3}}
                              {{3},{2,3},{1,2,3}}
                              {{1,3},{2,3},{1,2,3}}
                              {{2},{3},{2,3},{1,2,3}}
                              {{2},{1,3},{2,3},{1,2,3}}
                              {{3},{1,3},{2,3},{1,2,3}}
                              {{1,2},{1,3},{2,3},{1,2,3}}
                              {{2},{3},{1,3},{2,3},{1,2,3}}
                              {{3},{1,2},{1,3},{2,3},{1,2,3}}
                              {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
(End)
		

References

• D. E. Knuth, The Art of Computer Programming, Vol. 4, Section 7.1.1

Links

• Daniel Borchmann, Bernhard Ganter, Concept Lattice Orbifolds - First Steps, Proceedings of the 7th International Conference on Formal Concept Analysis (ICFCA 2009), 22-37 (Reference points to A108799).
• G. Brinkmann and R. Deklerck, Generation of Union-Closed Sets and Moore Families, arXiv:1701.03751 [math.CO], 2017.
• G. Brinkmann and R. Deklerck, Generation of Union-Closed Sets and Moore Families, Journal of Integer Sequences, Vol.21 (2018), Article 18.1.7.
• P. Colomb, A. Irlande and O. Raynaud, Counting of Moore Families for n=7, International Conference on Formal Concept Analysis (2010).

Crossrefs

Cf. A102894, A102895, A102897.

The labeled case is A102896.

The covering case is A108798.

The same for intersection instead of union is A108800.

The case with empty edges allowed is A193675.

Cf. A000612, A001930, A003180, A306445, A326875, A326883.

Formula

a(n) = A193675(n)/2.

Extensions

a(6) received Aug 17 2005

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

a(7) from Gunnar Brinkmann, Feb 07 2018