## 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 $P\left(\pi \right)=\left\{i\in \left\{1,2,\dots ,n\right\}\mid \text{}i\text{isapeakof}\pi \text{}\right\}$. 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 ${\pi}_{1}{\pi}_{2}\cdots {\pi}_{n}\phantom{\rule{0.3em}{0ex}}|\phantom{\rule{0.3em}{0ex}}{\pi}_{n+1}\cdots {\pi}_{2n}$ 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.

## Citation

Alexander Diaz-Lopez. Pamela E. Harris. Erik Insko. Darleen Perez-Lavin. "Peak sets of classical Coxeter groups." Involve 10 (2) 263 - 290, 2017. https://doi.org/10.2140/involve.2017.10.263

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