A335330 Triangle read by rows: T(n,k) is the number of k-colored graphs on n nodes with restricted labels, n>=0, 0<=k<=n.

1, 0, 1, 0, 1, 2, 0, 1, 8, 8, 0, 1, 32, 96, 64, 0, 1, 160, 1152, 2048, 1024, 0, 1, 1088, 17920, 65536, 81920, 32768, 0, 1, 10368, 399360, 2752512, 6553600, 6291456, 2097152, 0, 1, 139520, 13393920, 168820736, 692060160, 1207959552, 939524096, 268435456

Offset: 0

Author

Geoffrey Critzer, Jun 01 2020

Keywords

• nonn
• tabl

Comments

Here, a k-colored graph on n nodes with restricted labels is a labeled k-colored graph (as in A046860) with color set {c1,c2,...,ck} such that the nodes assigned to color c1 are labeled with the integers {1,2,...,n_c1}, the nodes assigned to color c2 are labeled with the next smallest n_c2 integers {n_c1+1,n_c1+2,... n_c1+n_c2}, and generally the nodes assigned to color cj are labeled with the smallest n_cj integers not previously used to label nodes having colors c1,c2,...c(j-1) where n_cj is the number of nodes having color cj and n_c1+n_c2+...+n_ck=n and each n_cj>0.

Examples

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,    2;
  0, 1,    8,     8;
  0, 1,   32,    96,    64;
  0, 1,  160,  1152,  2048,  1024;
  0, 1, 1088, 17920, 65536, 81920, 32768;
  ...
		

Crossrefs

Row sums give: A335343.

Cf. A046860, A006125 (main diagonal).

Programs

Mathematica

nn = 6; e[x_] := Sum[x^n/2^Binomial[n, 2],{n,0,nn}];Table[Take[(Table[2^Binomial[n, 2], {n, 0, nn}] CoefficientList[Series[1/(1 - y (e[x] - 1)), {x, 0, nn}], {x, y}])[[i]],i], {i, 1, nn + 1}] // Grid

Formula

Let E(x) = Sum_{n>=0} x^n/2^C(n,2). Then 1/(1-y(E(x)-1)) = Sum_{n>=0} Sum_{k=0..n} T(n,k) y^k*x^n/2^C(n,2).