A326868 - OEIS

%I #12 Oct 28 2023 12:07:54

%S 1,1,4,64,6048,8064000,1196002238976

%N Number of connected connectedness systems on n vertices.

%C 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 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 a(n > 1) = 2^n * A072447(n).

%F Logarithmic transform of A326870.

%e The a(3) = 64 connected connectedness systems:

%e   {{123}}              {{1}{123}}

%e   {{12}{123}}          {{2}{123}}

%e   {{13}{123}}          {{3}{123}}

%e   {{23}{123}}          {{1}{12}{123}}

%e   {{12}{13}{123}}      {{1}{13}{123}}

%e   {{12}{23}{123}}      {{1}{23}{123}}

%e   {{13}{23}{123}}      {{2}{12}{123}}

%e   {{12}{13}{23}{123}}  {{2}{13}{123}}

%e                        {{2}{23}{123}}

%e                        {{3}{12}{123}}

%e                        {{3}{13}{123}}

%e                        {{3}{23}{123}}

%e                        {{1}{12}{13}{123}}

%e                        {{1}{12}{23}{123}}

%e                        {{1}{13}{23}{123}}

%e                        {{2}{12}{13}{123}}

%e                        {{2}{12}{23}{123}}

%e                        {{2}{13}{23}{123}}

%e                        {{3}{12}{13}{123}}

%e                        {{3}{12}{23}{123}}

%e                        {{3}{13}{23}{123}}

%e                        {{1}{12}{13}{23}{123}}

%e                        {{2}{12}{13}{23}{123}}

%e                        {{3}{12}{13}{23}{123}}

%e .

%e   {{1}{2}{123}}              {{1}{2}{3}{123}}

%e   {{1}{3}{123}}              {{1}{2}{3}{12}{123}}

%e   {{2}{3}{123}}              {{1}{2}{3}{13}{123}}

%e   {{1}{2}{12}{123}}          {{1}{2}{3}{23}{123}}

%e   {{1}{2}{13}{123}}          {{1}{2}{3}{12}{13}{123}}

%e   {{1}{2}{23}{123}}          {{1}{2}{3}{12}{23}{123}}

%e   {{1}{3}{12}{123}}          {{1}{2}{3}{13}{23}{123}}

%e   {{1}{3}{13}{123}}          {{1}{2}{3}{12}{13}{23}{123}}

%e   {{1}{3}{23}{123}}

%e   {{2}{3}{12}{123}}

%e   {{2}{3}{13}{123}}

%e   {{2}{3}{23}{123}}

%e   {{1}{2}{12}{13}{123}}

%e   {{1}{2}{12}{23}{123}}

%e   {{1}{2}{13}{23}{123}}

%e   {{1}{3}{12}{13}{123}}

%e   {{1}{3}{12}{23}{123}}

%e   {{1}{3}{13}{23}{123}}

%e   {{2}{3}{12}{13}{123}}

%e   {{2}{3}{12}{23}{123}}

%e   {{2}{3}{13}{23}{123}}

%e   {{1}{2}{12}{13}{23}{123}}

%e   {{1}{3}{12}{13}{23}{123}}

%e   {{2}{3}{12}{13}{23}{123}}

%t Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],n==0||MemberQ[#,Range[n]]&&SubsetQ[#,Union@@@Select[Tuples[#,2],Intersection@@#!={}&]]&]],{n,0,4}]

%Y The case without singletons is A072447.

%Y The not necessarily connected case is A326866.

%Y The unlabeled case is A326869.

%Y The BII-numbers of these set-systems are A326879.

%Y Cf. A072445, A072446, A102896, A306445, A323818, A326867, A326870, A326872.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Jul 29 2019

%E a(6) corrected by _Christian Sievers_, Oct 28 2023