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 $φ_{FZ}$ which takes permutations to restricted Laguerre histories, we show that permutations $π$ and $f_A(π)$ 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 $D(π)$ when one samples $π$ from $\mathfrak{S}_n$ at random. As an application of our results, we get the mean and variance of the statistic $\mathrm{drops}(π)$ when we sample $π$ from $\mathcal{A}_n$ at random.

Figures (2)

• Figure 1: Intermediate permutations.
• Figure 2: The Bruhat order on $\mathfrak{S}_4$, taken from page 31 of Bjorner and Brenti, bjorner-brenti.

Theorems & Definitions (57)

• Theorem 1: Reifegerste, Petersen and Tenner
• Theorem 2: Sivasubramanian
• Theorem 3
• Corollary 4
• Example 5
• Remark 6
• Remark 7
• Remark 8
• Example 9
• Example 10