cp's OEIS Frontend

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.

%I A072444 #14 Oct 28 2023 15:27:52
%S A072444 1,1,2,6,47,3095,26897732
%N 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.
%C A072444 From _Gus Wiseman_, Aug 01 2019: (Start)
%C A072444 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:
%C A072444   {}
%C A072444   {{1,2}}
%C A072444   {{1,2,3}}
%C A072444   {{2,3},{1,2,3}}
%C A072444   {{1,3},{2,3},{1,2,3}}
%C A072444   {{1,2},{1,3},{2,3},{1,2,3}}
%C A072444 (End)
%H A072444 Wim van Dam, <a href="http://www.cs.berkeley.edu/~vandam/subpowersets/sequences.html">Sub Power Set Sequences</a>
%F A072444 Euler transform of A072445. - _Andrew Howroyd_, Oct 28 2023
%e A072444 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}}.
%Y A072444 The connected case is A072445.
%Y A072444 The labeled case is A072446.
%Y A072444 Unlabeled set-systems closed under union are A193674.
%Y A072444 Unlabeled connectedness systems are A326867.
%Y A072444 Cf. A072447, A092918, A326866, A326871, A326873.
%K A072444 nonn,more
%O A072444 0,3
%A A072444 Wim van Dam (vandam(AT)cs.berkeley.edu), Jun 18 2002
%E A072444 a(0)=1 prepended and a(6) corrected by _Andrew Howroyd_, Oct 28 2023