#### Vol. 10, No. 2, 2017

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Peak sets of classical Coxeter groups

### Alexander Diaz-Lopez, Pamela E. Harris, Erik Insko and Darleen Perez-Lavin

Vol. 10 (2017), No. 2, 263–290
##### 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 . 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 $|P\left(S;n\right)|=p\left(n\right){2}^{n-|S|-1}$, 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 $|{P}_{B}\left(S;n\right)|=p\left(n\right){2}^{2n-|S|-1}$, 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.

##### Keywords
binomial coefficient, peak, permutation, signed permutation, permutation pattern
##### Mathematical Subject Classification 2010
Primary: 05A05, 05A10, 05A15
##### Milestones 