A bijective proof of an unusual symmetric group generating function

Abstract

For $σ\in S_n$, let $D(σ) = \{i : σ_{i} > σ_{i+1}\}$ denote the descent set of $σ$. The length of the permutation is the number of inversions, denoted by $inv(σ) = \big | \{(i,j) : i<j, σ_i > σ_j\} \big |$. Define an unusual quadratic statisitic by $baj(σ) = \sum_{i \in D(σ)} i (n-i)$. We present here a bijective proof of the identity $\sum_{{σ\in S_n} \atop {σ(n) = k}} q^{baj(σ) - inv(σ)} = \prod_{i=1}^{n-1} {1-q^{i (n-i)} \over {1-q^i}}$ where $k$ is a fixed integer.

4 pages