A072444 - OEIS

%I #14 Oct 28 2023 15:27:52

%S 1,1,2,6,47,3095,26897732

%N 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 From _Gus Wiseman_, Aug 01 2019: (Start)

%C 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   {}

%C   {{1,2}}

%C   {{1,2,3}}

%C   {{2,3},{1,2,3}}

%C   {{1,3},{2,3},{1,2,3}}

%C   {{1,2},{1,3},{2,3},{1,2,3}}

%C (End)

%H Wim van Dam, <a href="http://www.cs.berkeley.edu/~vandam/subpowersets/sequences.html">Sub Power Set Sequences</a>

%F Euler transform of A072445. - _Andrew Howroyd_, Oct 28 2023

%e 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 The connected case is A072445.

%Y The labeled case is A072446.

%Y Unlabeled set-systems closed under union are A193674.

%Y Unlabeled connectedness systems are A326867.

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

%K nonn,more

%O 0,3

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

%E a(0)=1 prepended and a(6) corrected by _Andrew Howroyd_, Oct 28 2023