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 A327112 Number of set-systems covering n vertices with cut-connectivity >= 2, or 2-cut-connected set-systems. 9
 0, 0, 4, 72, 29856 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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). LINKS EXAMPLE Non-isomorphic representatives of the a(3) = 72 set-systems:   {{123}}   {{3}{123}}   {{23}{123}}   {{2}{3}{123}}   {{1}{23}{123}}   {{3}{23}{123}}   {{12}{13}{23}}   {{13}{23}{123}}   {{1}{2}{3}{123}}   {{1}{3}{23}{123}}   {{2}{3}{23}{123}}   {{3}{12}{13}{23}}   {{2}{13}{23}{123}}   {{3}{13}{23}{123}}   {{12}{13}{23}{123}}   {{1}{2}{3}{23}{123}}   {{2}{3}{12}{13}{23}}   {{1}{2}{13}{23}{123}}   {{2}{3}{13}{23}{123}}   {{3}{12}{13}{23}{123}}   {{1}{2}{3}{12}{13}{23}}   {{1}{2}{3}{13}{23}{123}}   {{2}{3}{12}{13}{23}{123}}   {{1}{2}{3}{12}{13}{23}{123}} MATHEMATICA 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]]]]]]]]]; 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]]]; Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], Union@@#==Range[n]&&vConn[#]>=2&]], {n, 0, 3}] CROSSREFS Covering 2-cut-connected graphs are A013922, if we assume A013922(2) = 1. Covering 1-cut-connected antichains (clutters) are A048143, if we assume A048143(0) = A048143(1) =0. Covering 2-cut-connected antichains (blobs) are A275307, if we assume A275307(1) = 0. Covering set-systems with cut-connectivity 2 are A327113. 2-vertex-connected integer partitions are A322387. BII-numbers of set-systems with cut-connectivity >= 2 are A327101. The cut-connectivity of the set-system with BII-number n is A326786(n). Cf. A002218, A003465, A013922, A259862, A323818, A327082, A327126, A327130. Sequence in context: A087315 A081460 A327040 * A284673 A055556 A168299 Adjacent sequences:  A327109 A327110 A327111 * A327113 A327114 A327115 KEYWORD nonn,more AUTHOR Gus Wiseman, Aug 24 2019 STATUS approved

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Last modified April 13 09:12 EDT 2021. Contains 342935 sequences. (Running on oeis4.)