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A327126
Triangle read by rows where T(n,k) is the number of labeled simple graphs covering n vertices with cut-connectivity k.
17
1, 0, 0, 0, 0, 1, 0, 3, 0, 1, 3, 28, 9, 0, 1, 40, 490, 212, 25, 0, 1, 745, 15336, 9600, 1692, 75, 0, 1
OFFSET
0,8
COMMENTS
We define the cut-connectivity of a graph to be the minimum number of vertices that must be removed (along with any incident edges) to obtain a disconnected or empty graph, with the exception that a graph with one vertex and no edges has cut-connectivity 1. Except for complete graphs, this is the same as vertex-connectivity.
EXAMPLE
Triangle begins:
1
0 0
0 0 1
0 3 0 1
3 28 9 0 1
40 490 212 25 0 1
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], {2}]], Union@@#==Range[n]&&cutConnSys[Range[n], #]==k&]], {n, 0, 4}, {k, 0, n}]
CROSSREFS
After the first column, same as A327125.
Column k = 0 is A327070.
Column k = 1 is A327114.
Row sums are A006129.
Different from A327069.
Row sums without the first column are A001187, if we assume A001187(0) = A001187(1) = 0.
Row sums without the first two columns are A013922.
Sequence in context: A353077 A115142 A346203 * A273083 A307451 A247629
KEYWORD
nonn,more,tabl
AUTHOR
Gus Wiseman, Aug 25 2019
EXTENSIONS
a(21)-a(27) from Robert Price, May 20 2021
STATUS
approved