The number of minimal length factorizations of a permutation into star transpositions. For a permutation $\pi\in\mathfrak S_n$ a minimal length factorization into star transpositions is a factorization of the form $$\pi = \tau_{i_1} \cdots \tau_{i_k}, 2 \leq i_1,\ldots,i_k \leq n,$$ where $\tau_a = (1,a)$ for $2 \leq a \leq n$ and $k$ is minimal. [1, lem.2.1] shows that the minimal length of such a factorization is $n+m-a-1$, where $m$ is the number of non-trival cycles not containing the element $1$, and $a$ is the number of fixed points different from $1$, see [[St001077]]. [2, cor.2] shows that the number of such minimal factorizations is $$ \frac{(n+m-2(k+1))!}{(n-k)!}\ell_1\cdots\ell_m, $$ where $\ell_1,\dots,\ell_m$ is the cycle type of $\pi$ and $k$ is the number of fixed point different from $1$.