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A327366 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and minimum vertex-degree k. 6
1, 1, 0, 1, 1, 0, 4, 3, 1, 0, 23, 31, 9, 1, 0, 256, 515, 227, 25, 1, 0, 5319, 15381, 10210, 1782, 75, 1, 0, 209868, 834491, 815867, 221130, 15564, 231, 1, 0, 15912975, 83016613, 116035801, 47818683, 5499165, 151455, 763, 1, 0, 2343052576, 15330074139, 29550173053, 18044889597, 3291232419, 158416629, 1635703, 2619, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

The minimum vertex-degree of the empty graph is infinity. It has been included here under k = 0. - Andrew Howroyd, Mar 09 2020

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..230 (rows n = 0..20)

Gus Wiseman, The graphs with 4 vertices and minimum vertex-degree k (row n = 4).

EXAMPLE

Triangle begins:

     1

     1     0

     1     1     0

     4     3     1     0

    23    31     9     1     0

   256   515   227    25     1     0

  5319 15381 10210  1782    75     1     0

MATHEMATICA

Table[Length[Select[Subsets[Subsets[Range[n], {2}]], k==If[#=={}||Union@@#!=Range[n], 0, Min@@Length/@Split[Sort[Join@@#]]]&]], {n, 0, 5}, {k, 0, n}]

PROG

(PARI)

GraphsByMaxDegree(n)={

  local(M=Map(Mat([x^0, 1])));

  my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));

  my(merge(r, p, v)=acc(p + sum(i=1, poldegree(p)-r-1, polcoef(p, i)*(1-x^i)), v));

  my(recurse(r, p, i, q, v, e)=if(i<0, merge(r, x^e+q, v), my(t=polcoef(p, i)); for(k=0, t, self()(r, p, i-1, (t-k+x*k)*x^i+q, binomial(t, k)*v, e+k))));

  for(k=2, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], my(p=src[i, 1]); recurse(n-k, p, poldegree(p), 0, src[i, 2], 0)));

  Mat(M);

}

Row(n)={if(n==0, [1], my(M=GraphsByMaxDegree(n), u=vector(n+1)); for(i=1, matsize(M)[1], u[n-poldegree(M[i, 1])]+=M[i, 2]); u)}

{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Mar 09 2020

CROSSREFS

Row sums are A006125.

Row sums without the first column are A006129.

Row sums without the first two columns are A100743.

Column k = 0 is A327367(n > 0).

Column k = 1 is A327227.

The unlabeled version is A294217.

Cf. A059167, A245797, A327069, A327103, A327334, A327369.

Sequence in context: A316656 A083904 A215861 * A327069 A327334 A195596

Adjacent sequences:  A327363 A327364 A327365 * A327367 A327368 A327369

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Sep 04 2019

EXTENSIONS

Terms a(28) and beyond from Andrew Howroyd, Sep 09 2019

STATUS

approved

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Last modified March 3 09:46 EST 2021. Contains 341760 sequences. (Running on oeis4.)