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A327149 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of simple labeled graphs covering n vertices with non-spanning edge-connectivity k. 8
1, 0, 1, 0, 0, 3, 1, 3, 12, 15, 10, 1, 40, 180, 297, 180, 60, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

The non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty graph.

LINKS

Table of n, a(n) for n=0..18.

FORMULA

A327148(n,k) = Sum_{m = 0..n} binomial(n,m) T(m,k). In words, column k is the inverse binomial transform of column k of A327148.

EXAMPLE

Triangle begins:

   1

   {}

   0   1

   0   0   3   1

   3  12  15  10   1

  40 180 297 180  60  10   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]]]]]]]]];

eConn[sys_]:=If[Length[csm[sys]]!=1, 0, Length[sys]-Max@@Length/@Select[Union[Subsets[sys]], Length[csm[#]]!=1&]];

Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Union@@#==Range[n]&&eConn[#]==k&]], {n, 0, 4}, {k, 0, Binomial[n, 2]}]//.{foe___, 0}:>{foe}

CROSSREFS

Row sums are A006129.

Column k = 0 is A327070.

Column k = 1 is A327079.

The corresponding triangle for vertex-connectivity is A327126.

The corresponding triangle for spanning edge-connectivity is A327069.

The non-covering version is A327148.

The unlabeled version is A327201.

Cf. A001187, A263296, A322338, A322395, A326787, A327097, A327099, A327102, A327125, A327129, A327144.

Sequence in context: A233168 A001351 A216021 * A173424 A143081 A179658

Adjacent sequences:  A327146 A327147 A327148 * A327150 A327151 A327152

KEYWORD

nonn,more,tabl

AUTHOR

Gus Wiseman, Aug 27 2019

STATUS

approved

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Last modified July 4 11:58 EDT 2020. Contains 335448 sequences. (Running on oeis4.)