Polya Enumeration Theorem cycle index variable interpretation

Question

The cycle index for a necklace with three beads up to rotations and no flips is $$P_G(x_1,x_2,x_3)=\frac{1}{3}(x_1^3+2x_3)$$ If we want to find how many such necklaces there are with four bead colors, we can do $$P_G(4,4,4)=\frac{1}{3}(4^3+2\cdot4)=24$$ If we want to find how many such necklaces there are with $n$ bead colors, we have $$P_G(n,n,n)=\frac{1}{3}(n^3+2n)$$ What is the interpretation when the variables are set to different values, i.e. what is the interpretation of $P_G(2,3,4)$.

Answer 1

Index cycle function (in your case $P_G(x_1,x_2,x_3)=\frac{1}{3}(x_1^3+2x_3)$) can be treated in several ways... In the most common situation (as you mentioned) you could use that like generating function:

Let say that you want color you object using $i_1, i_2, ..., i_k$ colors (let say the colors are $a_1, ... , a_k$). Then you can note that all items in the same cycle should has the same color so $$ x_t = a_1^{t} + \cdots + a_{k}^t $$ because in that cycle all items should be of color $a_1$ or all items of color $a_2$ or ... all items of color $a_k$. Then your result is coefficient with $a_1^{i_1}\cdot...\cdot a_k^{i_k}$


But if you write $P_G(2,3,4)$ you say that you want color all $1-$length cycles in use of chosen $2$ colors, all $2-$length cycles in use of chosen $3$ colors and all $3-$length cycles in use of chosen $4$ colors. I don't know how can exercises related with that problem looks like but it is possible to count that situation.