The number of parabolic double cosets with minimal element being the given permutation. For $w \in S_n$, this is $$\big| W_I \tau W_J\ :\ \tau \in S_n,\ I,J \subseteq S,\ w = \min\{W_I \tau W_J\}\big|$$ where $S$ is the set of simple transpositions, $W_K$ is the parabolic subgroup generated by $K \subseteq S$, and $\min\{W_I \tau W_J\}$ is the unique minimal element in weak order in the double coset $W_I \tau W_J$. [1] contains a combinatorial description of these parabolic double cosets which can be used to compute this statistic.