An identity of permutation statistics on the hyperoctahedral group

We define a statistic on the hyperoctahedral group B_n; this is a special case of one introduced by Stembridge and Waugh.  We show that a certain two-variable generating function involving this statistic factors into the product of a generating function over the symmetric group S_n and some simple binomials.  Then we mention related results by Macdonald and Carnevale-Shechter-Voll, and discuss what the correct way to state and prove our identity should be.  

Our motivation for this work comes from counting finite-index subgroups of the centrally amalgamated product of n copies of the discrete Heisenberg group whose profinite completion is isomorphic to that of the ambient group; we describe how the above-mentioned identity enables us to complete a computation begun 25 years ago by du Sautoy and Lubotzky.  

This is joint work with Mark Berman.