A072445 Number of subsets S of the power set P{1,2,...,n} such that: {1}, {2},..., {n} are all elements of S; {1,2,...,n} is an element of S; if X and Y are elements of S and X and Y have a nonempty intersection, then the union of X and Y is an element of S. The sets S are counted modulo permutations on the elements 1,2,...,n.

1, 1, 1, 4, 40, 3044, 26894586

Offset: 0

Author

Wim van Dam (vandam(AT)cs.berkeley.edu), Jun 18 2002

Keywords

• nonn
• more

Comments

We define a connectedness system to be a set of finite nonempty sets (edges) that is closed under taking the union of any two overlapping edges. It is connected if it is empty or contains an edge with all the vertices. Then a(n) is the number of unlabeled connected connectedness systems without singletons on n vertices. - Gus Wiseman, Aug 01 2019

Examples

			a(3) = 4 because of the 4 sets: {{1}, {2}, {3}, {1, 2, 3}}; {{1}, {2}, {3}, {1, 2}, {1, 2, 3}}; {{1}, {2}, {3}, {1, 2}, {1, 3}, {1, 2, 3}}; {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}.
		

Links

• Wim van Dam, Sub Power Set Sequences
• Gus Wiseman, Non-isomorphic representatives of the a(4) = 40 connected connectedness systems without singletons.

Crossrefs

The non-connected case is A072444.

The labeled case is A072447.

The case with singletons is A326869.

Cf. A072446, A092918, A108798, A326866, A326867, A326871, A326879.

Formula

Inverse Euler transform of A072444. - Andrew Howroyd, Oct 28 2023

Extensions

a(0)=1 prepended and a(6) corrected by Andrew Howroyd, Oct 28 2023