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Number of unlabeled connectedness systems covering n vertices.
7

%I #24 Jan 18 2024 04:41:35

%S 1,1,4,24,436,80460,1689114556

%N Number of unlabeled connectedness systems covering 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 covering if every vertex belongs to some edge.

%e Non-isomorphic representatives of the a(0) = 1 through a(3) = 24 connectedness systems:

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

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

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

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

%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 non-covering case without singletons is A072444.

%Y The case without singletons is A326899.

%Y First differences of A326867 (the non-covering case).

%Y Euler transform of A326869 (the connected case).

%Y The labeled case is A326870.

%Y Cf. A072445, A072446, A072447, A193674, A323818, A326866, A326868.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Jul 29 2019

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

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