A bijective proof of an unusual symmetric group generating function

Abstract

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 $inv(\sigma )=|\{(i,j):i<j,{\sigma }_i>{\sigma }_j\}|$. Define an unusual quadratic statisitic by $baj(\sigma )={\sum }_{i\in D(\sigma )}i(n-i)$. We present here a bijective proof of the identity ${\sum }_{\genfrac{}{}{0ex}{}{\sigma \in S_n}{\sigma (n)=k}}{q}^{baj(\sigma )-inv(\sigma )}={\prod }_{i=1}^{n-1}\frac{1-q^{i(n-i)}}{1-q^i}$ where $k$ is a fixed integer.

4 pages