cp's OEIS Frontend

A326866 Number of connectedness systems on n vertices.

%I A326866 #12 Oct 27 2023 23:39:41
%S A326866 1,2,8,96,6720,8130432,1196099819520
%N A326866 Number of connectedness systems on n vertices.
%C A326866 We define a connectedness system (investigated by Vim van Dam in 2002) to be a set of finite nonempty sets (edges) that is closed under taking the union of two overlapping edges.
%H A326866 Gus Wiseman, <a href="http://www.mathematica-journal.com/2017/12/every-clutter-is-a-tree-of-blobs/">Every Clutter Is a Tree of Blobs</a>, The Mathematica Journal, Vol. 19, 2017.
%F A326866 a(n) = 2^n * A072446(n).
%e A326866 The a(0) = 1 through a(2) = 8 connectedness systems:
%e A326866   {}  {}     {}
%e A326866       {{1}}  {{1}}
%e A326866              {{2}}
%e A326866              {{1,2}}
%e A326866              {{1},{2}}
%e A326866              {{1},{1,2}}
%e A326866              {{2},{1,2}}
%e A326866              {{1},{2},{1,2}}
%t A326866 Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],SubsetQ[#,Union@@@Select[Tuples[#,2],Intersection@@#!={}&]]&]],{n,0,3}]
%Y A326866 The case without singletons is A072446.
%Y A326866 The unlabeled case is A326867.
%Y A326866 The connected case is A326868.
%Y A326866 Binomial transform of A326870 (the covering case).
%Y A326866 The BII-numbers of these set-systems are A326872.
%Y A326866 Cf. A072444, A072447, A102896, A306445.
%K A326866 nonn,more
%O A326866 0,2
%A A326866 _Gus Wiseman_, Jul 29 2019
%E A326866 a(6) corrected by _Christian Sievers_, Oct 26 2023