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Number of unlabeled sets of subsets of {1..n} where every covered vertex is the unique common element of some subset of the edges.
10

%I #6 Aug 13 2019 13:19:02

%S 2,4,8,40,2464

%N Number of unlabeled sets of subsets of {1..n} where every covered vertex is the unique common element of some subset of the edges.

%C Alternatively, these are unlabeled sets of subsets of {1..n} whose dual is a (strict) antichain, also called T_1 sets of subsets. The dual of a set of subsets has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. An antichain is a set of subsets where no edge is a subset of any other.

%F a(n) = 2 * A326972(n).

%F a(n) = Sum_{k = 0..n} A327011(k).

%e Non-isomorphic representatives of the a(0) = 2 through a(2) = 8 sets of subsets:

%e {} {} {}

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

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

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

%e {{1},{2}}

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

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

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

%Y Unlabeled sets of subsets are A003180.

%Y Unlabeled T_0 sets of subsets are A326949.

%Y The labeled version is A326967.

%Y The case without empty edges is A326972.

%Y The covering case is A327011 (first differences).

%Y Cf. A003181, A059052, A326960, A326965, A326969, A326974, A326976.

%K nonn,more

%O 0,1

%A _Gus Wiseman_, Aug 13 2019