TITLE: The number of reduced expressions for the longest element of $B_n$?
QUESTION [4 upvotes]: Let $W=W_{\Phi}$ be a reflection group, with root system $\Phi$, and $\Delta=\{\alpha_1, ...,\alpha_n\}\subseteq \Phi$ a simple system. So $W$ is generated by the $s_{\alpha_i}=s_i$ for $i=1,2,...n. $ We know the fact that the length of the longest element $w_0$ of $W$ is $\mid\Phi^+\mid$. For example, in the type $A_3$, $w_0=s_2s_1s_3s_2s_1s_3$ is a longest element and its length is $6=\mid\Phi^+\mid$. You can see also What are the length of the longest element in a Coexter group for every type?
We know the fact that the number of reduced expressions for $w_0$ in type $A_n$ is given by 
\begin{align}
\frac{(\frac{1}{2}n(n+1))!}{r}
\end{align}
where $r$ is the product of the lengths of the hooks in the Young diagram corresponding to the partition $(n,n-1,n-2,...,2,1)$ of $\frac{1}{2}n(n+1)$.
My question is what is the number of reduced expressions for the longest element of $B_n$? Maybe this question is still an open question?

REPLY [1 votes]: The reduced words for the long element in B_n are in bijection with standard Young tableaux of shifted trapezoid shape $(2n-1,2n-3,\dots,1)$ . The first proof of this fact is in "Reduced decompositions in hyperoctahedral groups" by Witold Kraskiewicz. See here or here for additional references. 
Such tableaux are also enumerated by a hook-length formula, which for this shape reduces to
$$
\frac{(n^2)!}{\left(\prod^{n}_{i=1} (2i-1)^{n-i} \right)\prod^{n-3}_{j=0}\left(\prod^{n-2 - j}_{k = 1} 2(j + 2k) \right) }.
$$
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