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A327197
Number of set-systems covering n vertices with cut-connectivity 1.
8
0, 1, 0, 24, 1984
OFFSET
0,4
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 resulting empty edges) to obtain in a disconnected or empty set-system. Except for cointersecting set-systems (A327040), this is the same as vertex-connectivity.
FORMULA
Inverse binomial transform of A327128.
EXAMPLE
The a(3) = 24 set-systems:
{12}{13} {1}{12}{13} {1}{2}{12}{13} {1}{2}{3}{12}{13}
{12}{23} {1}{12}{23} {1}{2}{12}{23} {1}{2}{3}{12}{23}
{13}{23} {1}{13}{23} {1}{2}{13}{23} {1}{2}{3}{13}{23}
{2}{12}{13} {1}{3}{12}{13}
{2}{12}{23} {1}{3}{12}{23}
{2}{13}{23} {1}{3}{13}{23}
{3}{12}{13} {2}{3}{12}{13}
{3}{12}{23} {2}{3}{12}{23}
{3}{13}{23} {2}{3}{13}{23}
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]]]]]]]]];
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}]], Union@@#==Range[n]&&cutConnSys[Range[n], #]==1&]], {n, 0, 3}]
CROSSREFS
The BII-numbers of these set-systems are A327098.
The same for cut-connectivity 2 is A327113.
The non-covering version is A327128.
Sequence in context: A262010 A267075 A263605 * A194472 A246602 A001501
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Sep 01 2019
STATUS
approved