Let's say we have a pentagon and we want to find the total number of colors fixed by a permutation $\phi$. If we have $k$ colors, then the total number of colorings fixed by a permutation $(1)(25)(34)$ is $k^3$, where 3 is the number of cycles in $(1)(25)(34)$.
My question: why the total number of colors fixed by a permutation $(1)(25)(34)$ is $k^3$?
Here's the picture for context:
The permutation $(1)(25)(34)$ means that the colors at vertices $2$ and $5$, and $3$ and $4$, are interchanged. If the coloring is fixed by this interchange, then vertices $2$ and $5$, and vertices $3$ and $4$, must have the same color. This means that to define a coloring which is fixed by this permutation we have to choose $3$ colors, one for the vertex $1$, one for the pair $2$ and $5$, and one for the pair $3$ and $4$. Since there are $k$ colors, there are $k \cdot k \cdot k = k^3$ possible choices.