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Number of connected set-systems covering a subset of {1..n}.
7

%I #8 Aug 12 2019 03:24:25

%S 1,2,7,112,32253,2147316942,9223372023968335715,

%T 170141183460469231667123699322514272668,

%U 5789604461865809771178549250434395393752402807429031284280914691514037561273

%N Number of connected set-systems covering a subset of {1..n}.

%C A set-system is a finite set of finite nonempty sets.

%F Binomial transform of A323818.

%e The a(0) = 1 through a(2) = 7 set-systems:

%e {} {} {}

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

%e {{2}}

%e {{1,2}}

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

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

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

%t csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];

%t Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Length[csm[#]]<=1&]],{n,0,4}]

%Y Covering sets of subsets are A000371.

%Y Connected graphs are A001187.

%Y The unlabeled version is A309667.

%Y The BII-numbers of connected set-systems are A326749.

%Y The covering case is A323818.

%Y Cf. A007718, A048143, A058891, A092918, A300913, A304716, A326866, A326948.

%K nonn

%O 0,2

%A _Gus Wiseman_, Aug 10 2019