A326867 - OEIS

%I #17 Oct 28 2023 23:54:55

%S 1,2,6,30,466,80926,1689195482

%N Number of unlabeled 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.

%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.

%e Non-isomorphic representatives of the a(0) = 1 through a(3) = 30 connectedness systems:

%e   {}  {}     {}               {}

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

%e              {{1,2}}          {{1,2}}

%e              {{1},{2}}        {{1},{2}}

%e              {{2},{1,2}}      {{1,2,3}}

%e              {{1},{2},{1,2}}  {{1},{2,3}}

%e                               {{2},{1,2}}

%e                               {{1},{2},{3}}

%e                               {{3},{1,2,3}}

%e                               {{1},{2},{1,2}}

%e                               {{1},{3},{2,3}}

%e                               {{2,3},{1,2,3}}

%e                               {{2},{3},{1,2,3}}

%e                               {{1},{2,3},{1,2,3}}

%e                               {{1},{2},{3},{2,3}}

%e                               {{3},{2,3},{1,2,3}}

%e                               {{1},{2},{3},{1,2,3}}

%e                               {{1,3},{2,3},{1,2,3}}

%e                               {{1},{3},{2,3},{1,2,3}}

%e                               {{2},{3},{2,3},{1,2,3}}

%e                               {{2},{1,3},{2,3},{1,2,3}}

%e                               {{3},{1,3},{2,3},{1,2,3}}

%e                               {{1,2},{1,3},{2,3},{1,2,3}}

%e                               {{1},{2},{3},{2,3},{1,2,3}}

%e                               {{1},{2},{1,3},{2,3},{1,2,3}}

%e                               {{2},{3},{1,3},{2,3},{1,2,3}}

%e                               {{3},{1,2},{1,3},{2,3},{1,2,3}}

%e                               {{1},{2},{3},{1,3},{2,3},{1,2,3}}

%e                               {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

%e                               {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

%Y The case without singletons is A072444.

%Y The labeled case is A326866.

%Y The connected case is A326869.

%Y Partial sums of A326871 (the covering case).

%Y Cf. A072445, A072446, A072447, A102896, A306445, A326870, A326872.

%K nonn,more

%O 0,2

%A _Gus Wiseman_, Jul 29 2019

%E a(5) from _Andrew Howroyd_, Aug 10 2019

%E a(6) from _Andrew Howroyd_, Oct 28 2023