Equivalence class of permutations based on cycle decomposition and their inverses

Question

An equivalence class of permutations has come up in my research, and I'm wondering if anybody knows if it's named or has been studied before. If so, I'd appreciate being pointed towards more information.

Specifically, two permutations are considered equivalent if they have the same cycle decomposition, up to inverses of the cycles. So, for example, the permutations:

$(123)(456) \equiv (132)(456) \equiv (123)(465) \equiv (132)(465)$

And generally, if the $\sigma_{i}$ are disjoint cycles, then all permutations

$\sigma_{1}^{\pm}\sigma_{2}^{\pm} \cdots \sigma_{k}^{\pm}$

are equivalent. As I said, if anybody has seen this before and could point me towards information about it, I'd be most appreciative. Thank you!

Answer 1

If $f(n)$ is the number of equivalence classes, then $$ \sum_{n\geq 0} f(n)\frac{x^n}{n!} = \frac{\exp\left(\frac x2+ \frac{x^2}{4}\right)}{\sqrt{1-x}}. $$ This goes back to Frobenius. It is also the number of distinct monomials in the expansion of the determinant of an $n\times n$ generic symmetric matrix. See Enumerative Combinatorics, vol. 2, Example 5.2.9. Exercise 5.18 is similar.