Consider the space of polynomials with complex coefficients $\mathbb{C}[x_1,x_2,\dots,x_n]$ and let $\sigma$ be a permutation of $\{1,2,\cdots, n\}$ that acts on this space via $\sigma(x_i)=x_{\sigma(i)}$. Is there a nice generating function for the dimension of the degree $d$ fixed point subspace?
My guess: consider the cycle decomposition of $\{1,2,\dots,n\}$ with respect to $\sigma$, viewed as orbits for the subgroup of the symmetric group generated by $\sigma$. Call this set of orbits $\mathcal{O}$
Let $\ell$ be the lcm of the sizes of the orbits in $\mathcal{O}$, and for $O\in \mathcal{O}$, let $\omega_{|O|}$ denote the $|O|$-th primitive root of unity. For reasons I can provide justification for if needed, I believe the generating function will be:
$$\frac{1}{\ell}\sum\limits^{\ell-1}_{k=0} \prod\limits_{O\in \mathcal{O}}\prod\limits^{|O|-1}_{i,j=0} \frac{1}{1-\omega^{ijk}_{|O|}z}$$$$\frac{1}{\ell}\sum\limits^{\ell-1}_{k=0} \prod\limits_{O\in \mathcal{O}}\prod\limits^{|O|-1}_{i,j=0} \frac{1}{1-\omega^{ijk}_{|O|}z^j}$$
(the sum is related to https://math.stackexchange.com/questions/981996/generating-functions-for-partitions-of-n-with-an-even-number-of-parts-and-odd-nu)
But even if this is right, I would really appreciate a reference to avoid having to prove it!
Thank you