Peak sets of classical Coxeter groups

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

doi:10.2140/involve.2017.10.263

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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

DOI: 10.2140/involve.2017.10.263

Abstract

We say a permutation $π = π_{1} π_{2} \dots π_{n}$ in the symmetric group $S_{n}$ has a peak at index $i$ if $π_{i - 1} < π_{i} > π_{i + 1}$ and we let $P (π) = {i \in {1, 2, \dots, n} ∣ i is a peak of π}$ . Given a set $S$ of positive integers, we let $P (S; n)$ denote the subset of $S_{n}$ consisting of all permutations $π$ where $P (π) = S$ . In 2013, Billey, Burdzy, and Sagan proved $| P (S; n) | = p (n) 2^{n - | S | - 1}$ , where $p (n)$ is a polynomial of degree $max (S) - 1$ . 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 $| P_{B} (S; n) | = p (n) 2^{2 n - | S | - 1}$ , where $p (n)$ is the same polynomial of degree $max (S) - 1$ . In this paper we partition the sets $P (S; n) \subset S_{n}$ 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 $S_{2 n}$ , we partition these groups into bundles of permutations $π_{1} π_{2} \dots π_{n} | π_{n + 1} \dots π_{2 n}$ such that $π_{1} π_{2} \dots π_{n}$ has the same relative order as some permutation $σ_{1} σ_{2} \dots σ_{n} \in S_{n}$ . 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

Vol. 10, No. 2, 2017