Consider the symmetric group on $n$ symbols, $S_n$. Then, is there a way/ function to count the number of elements with a given number of transposition decomposition?
By transposition decomposition, I mean writing a permutation as a product of transpositions. So, for example, all transpositions have only one transposition factor, all $3$ cycles have exactly two transposition factors. But, the product of two disjoint transpositions also have a decomposition into two transposition factors. I think that counting the number of different cycle decompositions will help a lot. But has this been studied before and is there a function for eveluating it? Thanks beforehand.