A bijective proof of an unusual symmetric group generating function

TL;DR

This paper provides a bijective proof of a symmetric group generating function involving a novel quadratic statistic, connecting descent sets and inversions with a product formula.

Contribution

It introduces a new quadratic statistic on permutations and proves a related generating function identity via a bijective approach.

Findings

Established a bijective proof for the generating function identity.

Connected the quadratic statistic with descent sets and inversions.

Derived a product formula involving the new statistic.

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 $inv(\sigma) = \big | \{(i,j) : i<j, \sigma_i > \sigma_j\} \big |$. 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 $\sum_{{\sigma \in S_n} \atop {\sigma(n) = k}} q^{baj(\sigma) - inv(\sigma)} = \prod_{i=1}^{n-1} {{1-q^{i (n-i)}} \over {1-q^i}}$ where $k$ is a fixed integer.