A bijective proof of an unusual symmetric group generating function

For $\sigma \in S_n$, let $D(\sigma )=\{i:{\sigma }_i>{\sigma }_{i+1}\}$ denote the descent set of $\sigma$. The length of the permutation is the number of inversions, denoted by $inv(\sigma) = \big | {(i,j) : i<j, \sigma_i > \sigma_j} \big |$\mathrm{.}Defineanunusualquadraticstatisiticby$baj(\sigma) = \sum_{i \in D(\sigma)} i (n-i)$. We present here a bijective proof of the identity $\sum_{{\sigma \in S_n} \atop {\sigma(n) = k}} q^{baj(\sigma) - inv(\sigma)} = \prod_{i=1}^{n-1} {{1-q^{i (n-i)}} \over {1-q^i}}$where$k$ is a fixed integer.