Is there any recurrence relation known for the number of reduced words of the longest element in $S_n$ (not commutation classes)?
Edit: Sorry for unaccepting the answer, but I realized that I really would like to have a recurrence relation in $n$, so I would like to express the number of reduced words of the longest element in $S_n$ in terms of the numbers reduced words of the longest elements in $S_k$ for $k < n$.