I have a graph with $n$ verices, and I want to compute the number of ways to color the graph (with no adjacent vertices having the same color) using anywhere between $1$ and $n$ colors. This number (including permutations of colors) should be given by the chromatic polynomial for the graph.
However, I would like to present this number as a fraction of all possible ways to color the graph (without a restriction on not coloring adjacent vertices with the same color). The number of unique ways to do this for any $k<=n$ is $k$ multichoose $n$, or $(k+n-1)$ choose $n$.
The issue then is that the chromatic polynomial counts color permutations, whereas the multiset does not.
How do we place these on equivalent terms? In other words, how to we exclude color permutations from the chromatic polynomial or incorporate color permutations into the multichoose number?