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%I A326869 #17 Oct 28 2023 23:55:05
%S A326869 1,1,3,20,406,79964,1689032658
%N A326869 Number of unlabeled connected connectedness systems on n vertices.
%C A326869 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 contains an edge with all the vertices.
%e A326869 Non-isomorphic representatives of the a(3) = 20 connected connectedness systems:
%e A326869 {{1,2,3}}
%e A326869 {{3},{1,2,3}}
%e A326869 {{2,3},{1,2,3}}
%e A326869 {{2},{3},{1,2,3}}
%e A326869 {{1},{2,3},{1,2,3}}
%e A326869 {{3},{2,3},{1,2,3}}
%e A326869 {{1},{2},{3},{1,2,3}}
%e A326869 {{1,3},{2,3},{1,2,3}}
%e A326869 {{1},{3},{2,3},{1,2,3}}
%e A326869 {{2},{3},{2,3},{1,2,3}}
%e A326869 {{2},{1,3},{2,3},{1,2,3}}
%e A326869 {{3},{1,3},{2,3},{1,2,3}}
%e A326869 {{1,2},{1,3},{2,3},{1,2,3}}
%e A326869 {{1},{2},{3},{2,3},{1,2,3}}
%e A326869 {{1},{2},{1,3},{2,3},{1,2,3}}
%e A326869 {{2},{3},{1,3},{2,3},{1,2,3}}
%e A326869 {{3},{1,2},{1,3},{2,3},{1,2,3}}
%e A326869 {{1},{2},{3},{1,3},{2,3},{1,2,3}}
%e A326869 {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
%e A326869 {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
%Y A326869 The case without singletons is A072445.
%Y A326869 Connected set-systems are A092918.
%Y A326869 The not necessarily connected case is A326867.
%Y A326869 The labeled case is A326868.
%Y A326869 Euler transform is A326871 (the covering case).
%Y A326869 Cf. A072444, A072446, A072447, A102896, A323818, A326866, A326870, A326879.
%K A326869 nonn,more
%O A326869 0,3
%A A326869 _Gus Wiseman_, Jul 29 2019
%E A326869 a(5) from _Andrew Howroyd_, Aug 16 2019
%E A326869 a(6) from _Andrew Howroyd_, Oct 28 2023