Peak sets of classical Coxeter groups

Abstract

We say a permutation $\pi ={\pi }_1{\pi }_2\cdots {\pi }_n$ in the symmetric group ${\mathfrak{S}}_n$ has a peak at index $i$ if ${\pi }_{i-1}<{\pi }_i>{\pi }_{i+1}$ and we let $P\left(\pi \right)=\left\{i\in \left\{1,2,\dots ,n\right\}\mid i\text{ is a peak of }\pi \right\}$. Given a set $S$ of positive integers, we let $P\left(S;n\right)$ denote the subset of ${\mathfrak{S}}_n$ consisting of all permutations $\pi$ where $P\left(\pi \right)=S$. In 2013, Billey, Burdzy, and Sagan proved $\left|P\left(S;n\right)\right|=p\left(n\right)2^{n-\left|S\right|-1}$, where $p\left(n\right)$ is a polynomial of degree $max\left(S\right)-1$. In 2014, Castro-Velez et al. considered the Coxeter group of type $B_n$ as the group of signed permutations on $n$ letters and showed that $\left|P_B\left(S;n\right)\right|=p\left(n\right)2^{2n-\left|S\right|-1}$, where $p\left(n\right)$ is the same polynomial of degree $max\left(S\right)-1$. In this paper we partition the sets $P\left(S;n\right)\subset {\mathfrak{S}}_n$ studied by Billey, Burdzy, and Sagan into subsets of permutations that end with an ascent to a fixed integer $k$ (or a descent to a fixed integer $k$) and provide polynomial formulas for the cardinalities of these subsets. After embedding the Coxeter groups of Lie types $C_n$ and $D_n$ into ${\mathfrak{S}}_{2n}$, we partition these groups into bundles of permutations ${\pi }_1{\pi }_2\cdots {\pi }_n|{\pi }_{n+1}\cdots {\pi }_{2n}$ such that ${\pi }_1{\pi }_2\cdots {\pi }_n$ has the same relative order as some permutation ${\sigma }_1{\sigma }_2\cdots {\sigma }_n\in {\mathfrak{S}}_n$. This allows us to count the number of permutations in types $C_n$ and $D_n$ with a given peak set $S$ by reducing the enumeration to calculations in the symmetric group and sums across the rows of Pascal’s triangle.