A329543 Number of colored digraphs on n nodes with 1 to n colors assigned in a fixed order according the node count.

1, 1, 7, 116, 8282, 2168384, 2395241200, 10025552678528, 170709896192664592, 11335779739243176963200, 3029239690552322424003098368

Offset: 0

Author

Peter Dolland, Nov 16 2019

Keywords

• nonn
• more

Comments

The values are just the row sums of the irregular triangle A328773 and for n>=1 of the regular triangle A329541.

Colors C_1,...,C_n are assigned to n nodes in the way that a_i >= a_(i+1) >= 0 for 1<=i

a(n) gives the number of digraphs (see A000273) without restrictions, where nodes of the same color are not differentiated.

Crossrefs

Cf. A000041, A000273 (digraphs with one color), A053763 (digraphs with n colors), A328773 (digraphs to a given color scheme), A329541.

Programs

PARI

\\ here C(p) computes A328773 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)], , 4))}
{ for(n=0, 10, print(vecsum(Row(n)))) }

Formula

a(n) = Sum_{i=1..A000041(n)} A328773(n,i).

a(n) = Sum_{i=1..n} A329541(n,i) for n>=1.

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