A330693 Regular subtriangle of A328773: T(n,k) is the number of the balanced colored digraphs on n nodes with k colors, 1 <= k <= n.

1, 3, 4, 16, 36, 64, 218, 1104, 2112, 4096, 9608, 90416, 266496, 528384, 1048576, 1540944, 30194176, 135032832, 269500416, 537919488, 1073741824

Offset: 1

Author

Peter Dolland, Dec 25 2019

Keywords

• nonn
• tabl
• more

Comments

The color count, k, is a measure of the differentiation of the nodes: k = 1 means no differentiation at all, so we have A000273. k = n means all nodes are taken to be different, and we have A053763. With increasing differentiation T(n,k) < T(n,k+1) for 1 <= k < n.

Examples

			The balanced color scheme [2,2,2] has with A330661 the index 5 in A328773. So there are T(6,3) = A328773(6,5) = 135032832 digraphs with 6 nodes two each in the same color.
Triangle T(n,k) begins:
        1;
        3,        4;
       16,       36,        64;
      218,     1104,      2112,      4096;
     9608,    90416,    266496,    528384,   1048576;
  1540944, 30194176, 135032832, 269500416, 537919488, 1073741824;
  ...
		

Crossrefs

Cf. A328773, A330661, A000273, A053763.

Programs

PARI

\\ here C(p) computes sequence value for given partition.
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) = {sum(i=2, #v, sum(j=1, i-1, 2*gcd(v[i], v[j]))) + sum(i=1, #v, v[i]-1)}
C(p)={((i, v)->if(i>#p, 2^edges(v), my(s=0); forpart(q=p[i], s+=permcount(q)*self()(i+1, concat(v, Vec(q)))); s/p[i]!))(1, [])}
Row(n)={apply(C, vecsort([Vecrev(p) | p<-partitions(n),p[#p]-p[1]<=1], , 4))}
{ for(n=1, 6, print(Row(n))) }

Formula

T(n,1) = A000273(n).

T(n,n) = A053763(n) = 2^(n^2 - n).

T(n,k) = A328773(n,A330661(n,k)).