A327197 Number of set-systems covering n vertices with cut-connectivity 1.

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.

Links

Table of n, a(n) for n=0..4.

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.

Cf. A003465, A052442, A052443, A259862, A323818, A326786, A327101, A327112, A327114, A327126, A327229.

Sequence in context: A262010 A267075 A263605 * A194472 A246602 A001501

Adjacent sequences: A327194 A327195 A327196 * A327198 A327199 A327200

Keyword

nonn,more

Author

Gus Wiseman, Sep 01 2019

Status

approved