A108798 Number of nonisomorphic systems enumerated by A102894; that is, the number of inequivalent closure operators in which the empty set is closed. Also, the number of union-closed sets with n elements that contain the universe and the empty set.

1, 1, 3, 14, 165, 14480, 108281182, 2796163091470050

Offset: 0

Author

Don Knuth, Jul 01 2005

Keywords

• nonn
• more

Comments

Also the number of unlabeled finite sets of subsets of {1..n} that contain {} and {1..n} and are closed under intersection. - Gus Wiseman, Aug 02 2019

Examples

			From _Gus Wiseman_, Aug 02 2019: (Start)
Non-isomorphic representatives of the a(0) = 1 through a(3) = 14 union-closed sets of sets:
  {}  {}{1}  {}{12}        {}{123}
             {}{2}{12}     {}{3}{123}
             {}{1}{2}{12}  {}{23}{123}
                           {}{1}{23}{123}
                           {}{3}{23}{123}
                           {}{13}{23}{123}
                           {}{2}{3}{23}{123}
                           {}{2}{13}{23}{123}
                           {}{3}{13}{23}{123}
                           {}{12}{13}{23}{123}
                           {}{2}{3}{13}{23}{123}
                           {}{3}{12}{13}{23}{123}
                           {}{2}{3}{12}{13}{23}{123}
                           {}{1}{2}{3}{12}{13}{23}{123}
(End)
		

Links

• Maria Paola Bonacina and Nachum Dershowitz, Canonical ground Horn theories, Lecture Notes in Computer Science 7797, 35-71 (2013).
• G. Brinkmann and R. Deklerck, Generation of Union-Closed Sets and Moore Families, Journal of Integer Sequences, Vol.21 (2018), Article 18.1.7.
• G. Brinkmann and R. Deklerck, Generation of Union-Closed Sets and Moore Families, arXiv:1701.03751 [math.CO], 2017.
• Christopher S. Flippen, Minimal Sets, Union-Closed Families, and Frankl's Conjecture, Master's thesis, Virginia Commonwealth Univ., 2023.

Crossrefs

The labeled version is A102894.

Cf. A000612, A001930, A003180, A102895, A102897, A108800, A193674, A193675, A326867, A326869, A326883.

Formula

a(n) = A108800(n)/2.

Extensions

a(6) added (using A193674) by N. J. A. Sloane, Aug 02 2011

Added a(7), and reference to union-closed sets. - Gunnar Brinkmann, Feb 05 2018