Colourings, weight functions, and the pattern inventory

Let a finite group $G$ act on a finite set $X$, and let $C$ be a finite set of colours. A colouring is a function $f:X\to C$.

The action of $G$ on $X$ induces an action on colourings by

$(g\cdot f)(x):=f(g^{-1}x)\mathrm{.}$

Now let $R$ be a commutative ring and let $w:C\to R$ be a weight function. The weight of a colouring $f$ is

$\mathrm{wt}(f):={\prod }_{x\in X}w(f(x))\mathrm{.}$

Because the induced action only permutes the positions $x\in X$, every two colourings in the same $G$-orbit have the same weight.

The pattern inventory is the orbit sum

${\mathcal{I}}_G(X,C;w):={\sum }_{O\in C^X\mathrm{/}G}\mathrm{wt}(O),$

where $\mathrm{wt}(O)$ denotes the common weight of the colourings in the orbit $O$. When every colour has weight $1$, this is just the number of colouring orbits.