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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
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).
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).
Sequence in context: A087315 A081460 A327040 * A284673 A055556 A168299
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Aug 24 2019
STATUS
approved