cp's OEIS Frontend

A327112 Number of set-systems covering n vertices with cut-connectivity >= 2, or 2-cut-connected set-systems.

%I A327112 #12 Sep 01 2019 22:03:36
%S A327112 0,0,4,72,29856
%N A327112 Number of set-systems covering n vertices with cut-connectivity >= 2, or 2-cut-connected set-systems.
%C A327112 A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. The cut-connectivity of a set-system is the minimum number of vertices that must be removed (along with any empty or duplicate edges) to obtain a disconnected or empty set-system. Except for cointersecting set-systems (A327040), this is the same as vertex-connectivity (A327334, A327051).
%e A327112 Non-isomorphic representatives of the a(3) = 72 set-systems:
%e A327112   {{123}}
%e A327112   {{3}{123}}
%e A327112   {{23}{123}}
%e A327112   {{2}{3}{123}}
%e A327112   {{1}{23}{123}}
%e A327112   {{3}{23}{123}}
%e A327112   {{12}{13}{23}}
%e A327112   {{13}{23}{123}}
%e A327112   {{1}{2}{3}{123}}
%e A327112   {{1}{3}{23}{123}}
%e A327112   {{2}{3}{23}{123}}
%e A327112   {{3}{12}{13}{23}}
%e A327112   {{2}{13}{23}{123}}
%e A327112   {{3}{13}{23}{123}}
%e A327112   {{12}{13}{23}{123}}
%e A327112   {{1}{2}{3}{23}{123}}
%e A327112   {{2}{3}{12}{13}{23}}
%e A327112   {{1}{2}{13}{23}{123}}
%e A327112   {{2}{3}{13}{23}{123}}
%e A327112   {{3}{12}{13}{23}{123}}
%e A327112   {{1}{2}{3}{12}{13}{23}}
%e A327112   {{1}{2}{3}{13}{23}{123}}
%e A327112   {{2}{3}{12}{13}{23}{123}}
%e A327112   {{1}{2}{3}{12}{13}{23}{123}}
%t A327112 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 A327112 vConn[sys_]:=If[Length[csm[sys]]!=1,0,Min@@Length/@Select[Subsets[Union@@sys],Function[del,Length[csm[DeleteCases[DeleteCases[sys,Alternatives@@del,{2}],{}]]]!=1]]];
%t A327112 Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&vConn[#]>=2&]],{n,0,3}]
%Y A327112 Covering 2-cut-connected graphs are A013922, if we assume A013922(2) = 1.
%Y A327112 Covering 1-cut-connected antichains (clutters) are A048143, if we assume A048143(0) = A048143(1) =0.
%Y A327112 Covering 2-cut-connected antichains (blobs) are A275307, if we assume A275307(1) = 0.
%Y A327112 Covering set-systems with cut-connectivity 2 are A327113.
%Y A327112 2-vertex-connected integer partitions are A322387.
%Y A327112 BII-numbers of set-systems with cut-connectivity >= 2 are A327101.
%Y A327112 The cut-connectivity of the set-system with BII-number n is A326786(n).
%Y A327112 Cf. A002218, A003465, A013922, A259862, A323818, A327082, A327126, A327130.
%K A327112 nonn,more
%O A327112 0,3
%A A327112 _Gus Wiseman_, Aug 24 2019