A bijective proof of an unusual symmetric group generating function

Details A bijective proof of an unusual symmetric group generating function A bijective proof of an unusual symmetric group generating function

2003-10-19

arxiv.org/abs/math/0310301v1

Mathematics

Combinatorics

4 pages

Scientific paper

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 $inv(\sigma) = \big | \{(i,j) : i \sigma_j\} \big |$. 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.

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