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A327072 Triangle read by rows where T(n,k) is the number of labeled simple connected graphs with n vertices and exactly k bridges. 7
1, 1, 0, 0, 1, 0, 1, 0, 3, 0, 10, 12, 0, 16, 0, 253, 200, 150, 0, 125, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

A bridge is an edge that, if removed without removing any incident vertices, disconnects the graph. Connected graphs with no bridges are counted by A095983 (2-edge-connected graphs).

Warning: In order to be consistent with A001187, we have treated the n = 0 and n = 1 cases in ways that are not consistent with A095983.

LINKS

Table of n, a(n) for n=0..20.

Gus Wiseman, The 10 + 12 + 16 graphs counted in row n = 4.

EXAMPLE

Triangle begins:

    1

    1   0

    0   1   0

    1   0   3   0

   10  12   0  16   0

  253 200 150   0 125   0

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]]]]]]]]];

Table[If[n<=1&&k==0, 1, Length[Select[Subsets[Subsets[Range[n], {2}]], Union@@#==Range[n]&&Length[csm[#]]==1&&Count[Table[Length[Union@@Delete[#, i]]<n||Length[csm[Delete[#, i]]]>1, {i, Length[#]}], True]==k&]]], {n, 0, 4}, {k, 0, n}]

CROSSREFS

Column k = 0 is A095983, if we assume A095983(0) = A095983(1) = 1.

Column k = 1 is A327073.

Column k = n - 1 is A000272.

Row sums are A001187.

The unlabeled version is A327077.

Row sums without the first column are A327071.

Cf. A001349, A007146, A052446, A054592, A059166, A322395, A327069, A327148.

Sequence in context: A119957 A028852 A319202 * A327377 A095200 A090460

Adjacent sequences:  A327069 A327070 A327071 * A327073 A327074 A327075

KEYWORD

nonn,more,tabl

AUTHOR

Gus Wiseman, Aug 24 2019

STATUS

approved

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Last modified July 10 00:25 EDT 2020. Contains 335570 sequences. (Running on oeis4.)