cp's OEIS Frontend

A326868 Number of connected connectedness systems on n vertices.

%I A326868 #12 Oct 28 2023 12:07:54
%S A326868 1,1,4,64,6048,8064000,1196002238976
%N A326868 Number of connected connectedness systems on n vertices.
%C A326868 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 any two overlapping edges. It is connected if it is empty or contains an edge with all the vertices.
%H A326868 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 A326868 a(n > 1) = 2^n * A072447(n).
%F A326868 Logarithmic transform of A326870.
%e A326868 The a(3) = 64 connected connectedness systems:
%e A326868   {{123}}              {{1}{123}}
%e A326868   {{12}{123}}          {{2}{123}}
%e A326868   {{13}{123}}          {{3}{123}}
%e A326868   {{23}{123}}          {{1}{12}{123}}
%e A326868   {{12}{13}{123}}      {{1}{13}{123}}
%e A326868   {{12}{23}{123}}      {{1}{23}{123}}
%e A326868   {{13}{23}{123}}      {{2}{12}{123}}
%e A326868   {{12}{13}{23}{123}}  {{2}{13}{123}}
%e A326868                        {{2}{23}{123}}
%e A326868                        {{3}{12}{123}}
%e A326868                        {{3}{13}{123}}
%e A326868                        {{3}{23}{123}}
%e A326868                        {{1}{12}{13}{123}}
%e A326868                        {{1}{12}{23}{123}}
%e A326868                        {{1}{13}{23}{123}}
%e A326868                        {{2}{12}{13}{123}}
%e A326868                        {{2}{12}{23}{123}}
%e A326868                        {{2}{13}{23}{123}}
%e A326868                        {{3}{12}{13}{123}}
%e A326868                        {{3}{12}{23}{123}}
%e A326868                        {{3}{13}{23}{123}}
%e A326868                        {{1}{12}{13}{23}{123}}
%e A326868                        {{2}{12}{13}{23}{123}}
%e A326868                        {{3}{12}{13}{23}{123}}
%e A326868 .
%e A326868   {{1}{2}{123}}              {{1}{2}{3}{123}}
%e A326868   {{1}{3}{123}}              {{1}{2}{3}{12}{123}}
%e A326868   {{2}{3}{123}}              {{1}{2}{3}{13}{123}}
%e A326868   {{1}{2}{12}{123}}          {{1}{2}{3}{23}{123}}
%e A326868   {{1}{2}{13}{123}}          {{1}{2}{3}{12}{13}{123}}
%e A326868   {{1}{2}{23}{123}}          {{1}{2}{3}{12}{23}{123}}
%e A326868   {{1}{3}{12}{123}}          {{1}{2}{3}{13}{23}{123}}
%e A326868   {{1}{3}{13}{123}}          {{1}{2}{3}{12}{13}{23}{123}}
%e A326868   {{1}{3}{23}{123}}
%e A326868   {{2}{3}{12}{123}}
%e A326868   {{2}{3}{13}{123}}
%e A326868   {{2}{3}{23}{123}}
%e A326868   {{1}{2}{12}{13}{123}}
%e A326868   {{1}{2}{12}{23}{123}}
%e A326868   {{1}{2}{13}{23}{123}}
%e A326868   {{1}{3}{12}{13}{123}}
%e A326868   {{1}{3}{12}{23}{123}}
%e A326868   {{1}{3}{13}{23}{123}}
%e A326868   {{2}{3}{12}{13}{123}}
%e A326868   {{2}{3}{12}{23}{123}}
%e A326868   {{2}{3}{13}{23}{123}}
%e A326868   {{1}{2}{12}{13}{23}{123}}
%e A326868   {{1}{3}{12}{13}{23}{123}}
%e A326868   {{2}{3}{12}{13}{23}{123}}
%t A326868 Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],n==0||MemberQ[#,Range[n]]&&SubsetQ[#,Union@@@Select[Tuples[#,2],Intersection@@#!={}&]]&]],{n,0,4}]
%Y A326868 The case without singletons is A072447.
%Y A326868 The not necessarily connected case is A326866.
%Y A326868 The unlabeled case is A326869.
%Y A326868 The BII-numbers of these set-systems are A326879.
%Y A326868 Cf. A072445, A072446, A102896, A306445, A323818, A326867, A326870, A326872.
%K A326868 nonn,more
%O A326868 0,3
%A A326868 _Gus Wiseman_, Jul 29 2019
%E A326868 a(6) corrected by _Christian Sievers_, Oct 28 2023