Canonical reduced words and signed descent length enumeration in Coxeter groups

Abstract

Reifegerste and independently, Petersen and Tenner studied a statistic $\mathrm{drops}()$ on permutations in $\mathfrak{S}_n$. Two other studied statistics on $\mathfrak{S}_n$ are $\mathrm{depth}$ and $\mathrm{exc}$. Using descents in ${\it canonical\ reduced\ words}$ of elements in $\mathfrak{S}_n$, we give an involution $f_A: \mathfrak{S}_n \mapsto \mathfrak{S}_n$ that leads to a neat formula for the signed trivariate enumerator of $\mathrm{drops},\mathrm{depth}, \mathrm{exc}$ in $\mathfrak{S}_n$. This gives a simple formula for the signed univariate drops enumerator in $\mathfrak{S}_n$. For the type-B Coxeter group $\mathfrak{B}_n$ as well, using similar techniques, we show analogous results. For the type D Coxeter group, we again get analogous results, but our proof is inductive. Under the famous Foata-Zeilberger bijection $\phi_{FZ}$ which takes permutations to restricted Laguerre histories, we show that permutations $\pi$ and $f_A(\pi)$ map to the same Motzkin path, but have different history components. Using the Foata-Zeilberger bijection, we also get a continued fraction for the generating function enumerating the pair of statistics $\mathrm{drops}$ and $\mathrm{MAD}$. Graham and Diaconis determined the mean and the variance of the Spearman metric of disarray $D(\pi)$ when one samples $\pi$ from $\mathfrak{S}_n$ at random. As an application of our results, we get the mean and variance of the statistic $\mathrm{drops}(\pi)$ when we sample $\pi$ from $\mathcal{A}_n$ at random.