TITLE: How many permutations are there at a given Cayley distance from the identity?
QUESTION [5 upvotes]: 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 in the unitary group.

REPLY [7 votes]: 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, 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.