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%I #10 Aug 12 2019 22:31:35
%S 1,1,2,36,19020,2010231696,9219217412568364176,
%T 170141181796805105960861096082778425120,
%U 57896044618658097536026644159052312977171804852352892309392604715987334365792
%N Number of set-systems covering n vertices where every vertex is the unique common element of some subset of the edges, also called covering T_1 set-systems.
%C Same as A059523 except with a(1) = 1 instead of 2.
%C Alternatively, these are set-systems covering n vertices whose dual is a (strict) antichain. A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. An antichain is a set of sets, none of which is a subset of any other.
%F Inverse binomial transform of A326965.
%e The a(3) = 36 set-systems:
%e {{1}{2}{3}} {{12}{13}{23}{123}} {{2}{3}{12}{13}{23}}
%e {{12}{13}{23}} {{1}{2}{3}{12}{13}} {{2}{3}{12}{13}{123}}
%e {{1}{2}{3}{12}} {{1}{2}{3}{12}{23}} {{2}{12}{13}{23}{123}}
%e {{1}{2}{3}{13}} {{1}{2}{3}{13}{23}} {{3}{12}{13}{23}{123}}
%e {{1}{2}{3}{23}} {{1}{2}{12}{13}{23}} {{1}{2}{3}{12}{13}{23}}
%e {{1}{2}{13}{23}} {{1}{2}{3}{12}{123}} {{1}{2}{3}{12}{13}{123}}
%e {{1}{2}{3}{123}} {{1}{2}{3}{13}{123}} {{1}{2}{3}{12}{23}{123}}
%e {{1}{3}{12}{23}} {{1}{2}{3}{23}{123}} {{1}{2}{3}{13}{23}{123}}
%e {{2}{3}{12}{13}} {{1}{3}{12}{13}{23}} {{1}{2}{12}{13}{23}{123}}
%e {{1}{12}{13}{23}} {{1}{2}{13}{23}{123}} {{1}{3}{12}{13}{23}{123}}
%e {{2}{12}{13}{23}} {{1}{3}{12}{23}{123}} {{2}{3}{12}{13}{23}{123}}
%e {{3}{12}{13}{23}} {{1}{12}{13}{23}{123}} {{1}{2}{3}{12}{13}{23}{123}}
%t tmQ[eds_]:=Union@@Select[Intersection@@@Rest[Subsets[eds]],Length[#]==1&]==Union@@eds;
%t Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&tmQ[#]&]],{n,0,3}]
%Y Covering set-systems are A003465.
%Y Covering T_0 set-systems are A059201.
%Y The version with empty edges allowed is A326960.
%Y The non-covering version is A326965.
%Y Covering set-systems whose dual is a weak antichain are A326970.
%Y The unlabeled version is A326974.
%Y The BII-numbers of T_1 set-systems are A326979.
%Y Cf. A058891, A059052, A059523, A323818, A326972, A326973, A326976, A326977.
%K nonn
%O 0,3
%A _Gus Wiseman_, Aug 12 2019
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