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A327128 Number of set-systems with n vertices whose edge-set has cut-connectivity 1. 4
0, 1, 2, 27, 2084 (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. We define the cut-connectivity (A326786, A327237, A327126) of a set-system to be the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a disconnected or empty set-system, with the exception that a set-system with one vertex has cut-connectivity 1. Except for cointersecting set-systems (A326853, A327039, A327040), this is the same as vertex-connectivity (A327334, A327051).
LINKS
FORMULA
Binomial transform of A327197.
MATHEMATICA
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
cutConnSys[vts_, eds_]:=If[Length[vts]==1, 1, Min@@Length/@Select[Subsets[vts], Function[del, csm[DeleteCases[DeleteCases[eds, Alternatives@@del, {2}], {}]]!={Complement[vts, del]}]]];
Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], cutConnSys[Union@@#, #]==1&]], {n, 0, 3}]
CROSSREFS
The covering version is A327197.
The BII-numbers of these set-systems are A327098.
Sequence in context: A015217 A320417 A113094 * A320669 A299116 A076113
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Sep 02 2019
STATUS
approved

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Last modified March 29 10:59 EDT 2024. Contains 371277 sequences. (Running on oeis4.)