I recently rediscovered the rather nice formula for the order of the centraliser of a permutation in the symmetric group and its realtionship with conjugacy classes. I wondered whether we could say anything interesting if we break down this formula as I will try to explain.
The following is very well known: Suppose that $\sigma \in \Sigma_n$ is a permutation. Then $\sigma$ has cycle type $(a_1,a_2,\dots a_n)$ if (when written in disjoint cycle notation) $\sigma$ consists of $a_1$ cycles of length 1, $a_2$ cycles of length 2 and so on. The order of the centraliser of $\sigma$ is given by the following famous formula: $$ \lvert C(\sigma) \rvert = \prod_i a_i! i^{a_i}. $$
By thinking of $\Sigma_n$ acting on itself by conjugation and applying Burnside's lemma it is easy to see that the expected value of the order of the centraliser, $$ \frac{1}{n!} \sum_{\sigma \in \Sigma_n} \lvert C(\sigma) \rvert $$ is just the number of conjugacy classes in $\Sigma_n$. This number is also known to be the number of integer partitions of the first $n$ integers.
Less well known: Define two new (but related) functions $D,E : \sigma_n \rightarrow \mathbb{N}$ as follows: $$ D(\sigma) = \prod_i a_i ! \text{ and } E(\sigma) = \prod_i i^{a_i}$$ where again $\sigma$ has cycle type $(a_1,a_2,\dots,a_n)$.
Is there an interesting group theoretical/combinatorial interpretation of these functions? Can one use this to calculate the expected value of $D(\sigma)$ or $E(\sigma)$ perhaps using a Burnside's lemma type argument?