Vol. 10, No. 2, 2017

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ISSN: 1944-4184 (e-only)
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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

We say a permutation π = π1π2πn in the symmetric group Sn has a peak at index i if πi1 < πi > πi+1 and we let P(π) = {i {1,2,,n} i is a peak of π}. Given a set S of positive integers, we let P(S;n) denote the subset of Sn consisting of all permutations π where P(π) = S. In 2013, Billey, Burdzy, and Sagan proved |P(S;n)| = p(n)2n|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 Bn as the group of signed permutations on n letters and showed that |PB(S;n)| = p(n)22n|S|1 , where p(n) is the same polynomial of degree  max(S)1. In this paper we partition the sets P(S;n) Sn 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 Cn and Dn into S2n, we partition these groups into bundles of permutations π1π2πn|πn+1π2n such that π1π2πn has the same relative order as some permutation σ1σ2σn Sn. This allows us to count the number of permutations in types Cn and Dn 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.

binomial coefficient, peak, permutation, signed permutation, permutation pattern
Mathematical Subject Classification 2010
Primary: 05A05, 05A10, 05A15
Received: 11 September 2015
Revised: 21 January 2016
Accepted: 7 February 2016
Published: 10 November 2016

Communicated by Stephan Garcia
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
United States
Darleen Perez-Lavin
Department of Mathematics
University of Kentucky
Lexington, KY 40506
United States