A072444 Number of subsets S of the power set P{1,2,...,n} such that: {1}, {2},..., {n} are all elements 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, 2, 6, 47, 3095, 26897732

Offset: 0

Author

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

Keywords

• nonn
• more

Comments

From Gus Wiseman, Aug 01 2019: (Start)

If 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, then a(n) is the number of unlabeled connectedness systems on n vertices without singleton edges. Non-isomorphic representatives of the a(3) = 6 connectedness systems without singletons are:

{}

{{1,2}}

{{1,2,3}}

{{2,3},{1,2,3}}

{{1,3},{2,3},{1,2,3}}

{{1,2},{1,3},{2,3},{1,2,3}}

(End)

Examples

			a(3) = 6 because of the 6 sets: {{1}, {2}, {3}}; {{1}, {2}, {3}, {1, 2}}; {{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

Crossrefs

The connected case is A072445.

The labeled case is A072446.

Unlabeled set-systems closed under union are A193674.

Unlabeled connectedness systems are A326867.

Cf. A072447, A092918, A326866, A326871, A326873.

Formula

Euler transform of A072445. - Andrew Howroyd, Oct 28 2023

Extensions

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