A270843 Number of nonisomorphic edge colorings of the Petersen graph with exactly n colors.

1, 394, 122601, 8510140, 210940745, 2524556538, 17167621086, 72787256640, 202996629360, 382918536000, 492133561920, 424994169600, 236107872000, 76281004800, 10897286400

Offset: 1

Author

Marko Riedel, Mar 24 2016

Keywords

• nonn
• easy
• fini
• full

Comments

This is zero when n is more than fifteen because only fifteen edges are available.

These are not colorings in the strict sense, since there is no requirement that adjacent edges have different colors. - N. J. A. Sloane, Mar 28 2016

The value for n=15 is 15!/120 because all orbits are the same size namely 120 (order of the symmetric group on five elements) when each of the 15 edges has a unique color. - Marko Riedel, Mar 28 2016

Links

• Math StackExchange, Edge colorings of the Petersen graph

Crossrefs

Cf. A270842, A063843.

Formula

Cycle index of the automorphisms acting on the edges is (1/120)*S[1]^15+(5/24)*S[2]^6*S[1]^3+(1/4)*S[4]^3*S[2]*S[1]+(1/6)*S[3]^5+(1/6)*S[3]*S[6]^2+(1/5)*S[5]^3.

Inclusion-exclusion yields a(n) = sum(C(n, q)*(-1)^q*A270842(n - q), q = 0 .. n)