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

0,3

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

Links

Table of n, a(n) for n=0..7.

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.

Formula

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

Example

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)

Crossrefs

The labeled version is A102894.

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

Sequence in context: A090897 A120459 A272368 * A260995 A220438 A250001

Adjacent sequences: A108795 A108796 A108797 * A108799 A108800 A108801

Keyword

nonn,more

Author

Don Knuth, Jul 01 2005

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

Status

approved