The prefix exchange distance of a permutation. This is the number of star transpositions needed to write a permutation. In symbols, for a permutation $\pi\in\mathfrak S_n$ this is $$\min\{ k \mid \pi = \tau_{i_1} \cdots \tau_{i_k}, 2 \leq i_1,\ldots,i_k \leq n\},$$ where $\tau_a = (1,a)$ for $2 \leq a \leq n$. [1, Lem. 2.1] shows that the this length 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 $1$. One may find in [2] explicit formulas for its generating function and a combinatorial proof that it is asymptotically normal.