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 $(-1)^{C_\sigma}$.

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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 $C_\sigma=0,1,2,3,4$.
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**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.}