Permutations $\sigma$ in the symmetric group $S\_n$ can be characterized by their Cayley distance $C\_\sigma$, being the minimal number of transpositions needed to convert $\{1,2,3,\ldots n\}$ into $\sigma$. The sign of the permutation is $(-1)^{C\_\sigma}$.


For example, when $\sigma=\{2, 3, 4, 5, 1\}$, one has $C\_\sigma=4$ and for $\sigma=\{1, 2, 3, 5, 4\}$ one has $C\_\sigma=1$. Of the $5!$ permutations in $S\_5$ there are, respectively, $1,10,35,50,24 $ with Cayley distance $C\_\sigma=0,1,2,3,4$.


**Question:** What is the general formula that counts the number of permutations at a given Cayley distance?


This question was motivated by my attempt to check [an integral formula](https://mathoverflow.net/a/331312/11260) in the unitary group.



How many permutations are there at a given Cayley distance from the identity?
The Cayley distance of a permutation is also known as its absolute length, as can be found out by supplying a few values at 
<https://findstat.org/StatisticFinder/Permutations>, which yields <https://findstat.org/St000216>. There, we also find that for a permutation in $\mathfrak S\_n$ with $k$ cycles it is simply $n-k$. This fact is, for example, Problem 5.6 in [1].


[1] *Petersen, T. Kyle*, [**Eulerian numbers**](http://dx.doi.org/10.1007/978-1-4939-3091-3), Birkhäuser Advanced Texts. Basler Lehrbücher. New York, NY: Birkhäuser/Springer (ISBN 978-1-4939-3090-6/hbk; 978-1-4939-3091-3/ebook). xviii, 456 p. (2015). [ZBL1337.05001](https://zbmath.org/?q=an:1337.05001).


