A369928 -id:A369928 - cp's OEIS frontend

A123551 Triangle read by rows: T(n,k) gives number of unlabeled graphs without endpoints on n nodes and k edges, (n >= 0, 0 <= k <= n(n-1)/2).

1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 2, 4, 3, 2, 1, 1, 1, 0, 0, 1, 1, 2, 6, 8, 13, 16, 13, 8, 5, 2, 1, 1, 1, 0, 0, 1, 1, 2, 6, 10, 22, 48, 76, 97, 102, 84, 60, 39, 20, 10, 5, 2, 1, 1, 1, 0, 0, 1, 1, 2, 6, 10, 25, 64, 152, 331, 617, 930, 1173, 1253, 1140

Offset: 0

N. J. A. Sloane, Nov 14 2006

			Triangle begins:
[0] 1;
[1] 1;
[2] 1, 0;
[3] 1, 0, 0, 1;
[4] 1, 0, 0, 1, 1, 1, 1;
[5] 1, 0, 0, 1, 1, 2, 4, 3,  2,  1,  1;
[6] 1, 0, 0, 1, 1, 2, 6, 8, 13, 16, 13, 8, 5, 2, 1, 1;
  ...

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.

R. W. Robinson, Rows 0 through 16, flattened

Row sums are A004110.

Cf. A008406, A240168, A369928 (labeled).

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
row(n) = {my(s=0); sum(k=0, n, forpart(p=k, s+=permcount(p) * edges(p, w->1+y^w) * y^(n-k)*polcoef(prod(i=1, #p, 1-x^p[i]), n-k)/k!)); Vecrev(s, binomial(n,2)+1)}
{ for(n=0, 6, print(row(n))) } \\ Andrew Howroyd, Feb 07 2024

T(n,k) = A008406(n,k) - A240168(n,k). - Andrew Howroyd, Apr 16 2021

A369931 Triangle read by rows: T(n,k) is the number of labeled simple graphs with n edges and k vertices and without endpoints or isolated vertices.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 6, 12, 0, 0, 0, 1, 85, 70, 0, 0, 0, 0, 100, 990, 465, 0, 0, 0, 0, 45, 2805, 11550, 3507, 0, 0, 0, 0, 10, 3595, 59990, 140420, 30016, 0, 0, 0, 0, 1, 2697, 147441, 1174670, 1802682, 286884, 0, 0, 0, 0, 0, 1335, 222516, 4710300, 22467312, 24556140, 3026655

Offset: 1

Andrew Howroyd, Feb 08 2024

T(n,k) is the number of traceless symmetric binary matrices with 2n 1's and k rows and at least two 1's in every row.

			Triangle begins:
  0;
  0, 0;
  0, 0, 1;
  0, 0, 0, 3;
  0, 0, 0, 6,  12;
  0, 0, 0, 1,  85,   70;
  0, 0, 0, 0, 100,  990,    465;
  0, 0, 0, 0,  45, 2805,  11550,    3507;
  0, 0, 0, 0,  10, 3595,  59990,  140420,   30016;
  0, 0, 0, 0,   1, 2697, 147441, 1174670, 1802682, 286884;
  ...
The T(3,3) = 1 matrix is:
  [0 1 1]
  [1 0 1]
  [1 1 0]
The T(4,4) = 3 matrices are:
  [0 0 1 1]  [0 1 0 1]  [0 1 1 0]
  [0 0 1 1]  [1 0 1 0]  [1 0 0 1]
  [1 1 0 0]  [0 1 0 1]  [1 0 0 1]
  [1 1 0 0]  [1 0 1 0]  [0 1 1 0]

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Row sums are A370059.
Column sums are A100743.
Main diagonal is A001205.
Cf. A369928, A369932 (unlabeled).

G(n)={my(A=x/exp(x*y + O(x*x^n))); exp(y*x^2/2 - x + O(x*x^n)) * sum(k=0, n, (1 + y + O(y*y^n))^binomial(k, 2)*A^k/k!)}
T(n)={my(r=Vec(substvec(serlaplace(G(n)), [x, y], [y, x]))); vector(#r-1, i, Vecrev(Pol(r[i+1]/y), i))}

T(n,k) = k!*[x^k][y^n] exp(y*x^2/2 - x) * Sum_{j>=0} (1 + y)^binomial(j, 2)*(x/exp(y*x))^j/j!.

cp's OEIS Frontend

A123551 Triangle read by rows: T(n,k) gives number of unlabeled graphs without endpoints on n nodes and k edges, (n >= 0, 0 <= k <= n(n-1)/2).

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