More chromatic hoonja-doonja!
Suppose you have a bunch of tokens of different colors and weights. X_1 colors of weight 1 tokens, X_2 colors of weight 2 tokens, etc.
Let Γ be a graph. A (weighted) b-coloring of Γ is an assignment to each vertex of a set of tokens with total weight b, such that adjacent vertices have no tokens in common. Let χ_Γ(X_1, … X_b) be the number of b-colorings of Γ. I made up this definition but I assume it’s in the literature somewhere.
First of all, χ_Γ(X_1, … X_b) is a polynomial.
Is this multivariable “chromatic polynomial” of any interest? Well, here’s one place it comes up. By a degree-b polynomial coloring of Γ we mean an assignment of a monic squarefree degree d polynomial in R[x] to each vertex of Γ, so that adjacent vertices are labeled with coprime polynomials. Now let U_b(Γ) be the manifold parametrizing degree-b colorings of Γ.
Then the Euler characteristic of U_b(Γ) is χ_Γ(-1,1,0,…0).
I worked this out via the same kind of Lefschetz computation as in the previous post, but once you get the answer, you can actually derive this as a corollary of Stanley’s theorem. And it was presumably already known.
By the way: let V_n be the free vector space spanned by the b-colorings of Γ where all the tokens have weight 1; these are called fractional colorings sometimes. Then S_n acts on V_n by permutation of colors. The character of this action is a class function on S_n. More precisely, it is
χ_Γ(X_1, … X_b)
where X_i is now interpreted as a class function on S_n, sending a permutation to the number of i-cycles in its cycle decomposition. Of course the real thing going on behind the scenes is that the V_n form a finitely generated FI-module.