A380197 Number of ways to choose a simple labeled graph on [n] and properly color the vertices using the minimum number of colors.

1, 1, 3, 25, 423, 16261, 1266843, 200830225, 65750156223, 42834021462061, 55174125327583923

Offset: 0

Author

Geoffrey Critzer, Jan 22 2025

Keywords

• nonn
• more

Comments

a(n) is the chromatic polynomial evaluated at the chromatic number of a graph summed over all simple labeled graphs on [n].

Links

• Eric Weisstein's World of Mathematics, Chromatic Number.
• Eric Weisstein's World of Mathematics, Chromatic Polynomial.

Crossrefs

Cf. A006125, A372920, A229048, A084268.

Programs

Mathematica

(* this code only returns a(0) - a(7) using precomputed data in Mathematica 12 *)
nog = {1, 2, 4, 11, 34, 156, 1044}; Prepend[Table[Total[n!/Table[GraphData[{n, i},AutomorphismCount"], {i, 1, nog[[n]]}]*Table[GraphData[{n, i},ChromaticPolynomial"][GraphData[{n, i}, "ChromaticNumber"]], {i, 1, nog[[n]]}]], {n, 1, 7}],1]

Extensions

a(8)-a(10) from Andrew Howroyd, Jan 22 2025