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

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Ethics Statement Editorial Login Author Index Coming Soon Contacts ISSN: 1944-4184 (e-only) ISSN: 1944-4176 (print) Other MSP Journals
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
Received: 11 September 2015
Revised: 21 January 2016
Accepted: 7 February 2016
Published: 10 November 2016

Communicated by Stephan Garcia
##### Authors
 Alexander Diaz-Lopez Department of Mathematics and Statistics Swarthmore College Swarthmore, PA 19081 United States Pamela E. Harris Department of Mathematics and Statistics Williams College Williamstown, MA 01267 United States Erik Insko Department of Mathematics Florida Gulf Coast University Fort Myers, FL 33965 %-6565 United States Darleen Perez-Lavin Department of Mathematics University of Kentucky Lexington, KY 40506 United States