Number of reduced decompositions of the longest element of the Weyl group

Around the time Fomin and I wrote this paper, Tao Kai Lam applied the technique to type $D_n$. It emerged that it was "natural" to weight a reduced decomposition $\rho$ by $2^{d(\rho)}$, where $d(\rho)$ is the number of simple reflections in $\rho$ that correspond to the $n-2$ "nonbranch nodes" in the Coxeter diagram for $D_n$. Using this weighting, there is a nice product formula for the number of weighted reduced decompositions of the longest element, which I unfortunately have forgotten. I hope someone can redo this work.

This is easy to do in SageMath. E.g. the following code

G = WeylGroup("F4")
w = G.long_element_hardcoded()
print(w)
rw = w.reduced_words() 
len(rw)

outputs 2144892. If you want to look at some of these reduced words just examine the list rw. To create a list for classical types of different rank do

res = {}
for n in range(2,5):
    G = WeylGroup(["A", n])
    w = G.long_element_hardcoded()
    print("Calculating rank ", n)
    res[n] = len(w.reduced_words())