A340027 Number of inequivalent vertex colorings of connected graphs on n unlabeled vertices.

1, 1, 2, 7, 50, 520, 10665, 400220, 29204589, 4143245857, 1146827743079, 619412332805088, 653237982066620540, 1346571060160843394520, 5432476352054378478159877, 42947950068987980977264834190, 666212968663987333085874313873428, 20301440661023158546856805172595805762

Offset: 0

Author

Andrew Howroyd, Jan 02 2021

Keywords

• nonn

Comments

Equivalence is up to permutation of the colors. Adjacent vertices may have the same color.

Examples

			a(3) = 7 because there are 2 connected graphs on 3 vertices. The complete graph K_3 can be coloring in 3 ways (111, 112, 123) and the path graph P_3 can be colored in 4 ways (111, 112, 121, 123).
		

Crossrefs

Cf. A340024, A340025, A340026.

Programs

PARI

\\ See links in A339645 for combinatorial species functions.
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
graphsCycleIndex(n)={my(s=0); forpart(p=n, s+=permcount(p) * 2^edges(p) * sMonomial(p)); s/n!}
graphsSeries(n)={sum(k=0, n, graphsCycleIndex(k)*x^k) + O(x*x^n)}
InequivalentColoringsSeq(1+sLog(graphsSeries(15)))