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Let $B_n$ denote the group of signed permutations on $n$ letters. Is there a good explanation or understandable way to see why $$ \sum_{w\in B_n}q^{\text{inv}(w)}=(2n)_q(2n-2)_q\cdots(2)_q? $$

I've been thinking about it on and off while reading through Taylor's Geometry of the Classical Groups, but don't understand why this identity holds. I appreciate any explanation. Thanks!

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    You are using lots of terms/notations not everyone understands. Meet us halfway.2012-02-16
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    @GerryMyerson Terribly sorry about that, I was unaware the notation is not standard. I will add those in soon.2012-02-16

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