Kauffman states without proof in "Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds" that these 2 expressions for the q-deformed factorial are equal:
$$ [n]! = \prod_{k=1}^n \frac{1 - A^{-4k}}{1 - A^{-4}} = \sum_{\sigma \in S_n} A^{-4\tau(\sigma)}$$
, where the sum on the RHS is over all permutations in the symmetric group ($S_n$) and $\tau(\sigma)$ is the minimum number of transpositions required to compose $\sigma$. My strategy was to label permutations by their number of disjoint cycles $k$ so that $\tau(\sigma) = n - k$; also, the number of ways to split a permutation of $n$ objects into $k$ cycles is a Stirling number of the 1st kind ($s_{nk}$), which have the rising factorial as their generating function:
$$ \sum_{\sigma \in S_n} A^{-4\tau(\sigma)} \quad = \sum_{k = 0}^n s_{nk} \left(A^{-4}\right)^{n-k}\quad = A^{-4n} \prod_{k = 1}^n \left(A^4 + k - 1 \right)$$
However, regardless of what I do I cannot reconcile the two product formulae, assuming my product is valid in the first place. Any help would be greatly appreciated. Thankyou.