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

Abstract

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.

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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

Information

Received: 11 September 2015; Revised: 21 January 2016; Accepted: 7 February 2016; Published: 2017
First available in Project Euclid: 13 December 2017

zbMATH: 1350.05003
MathSciNet: MR3574301
Digital Object Identifier: 10.2140/involve.2017.10.263

Subjects:
Primary: 05A05 , 05A10 , 05A15

Keywords: binomial coefficient , peak , permutation , permutation pattern , signed permutation

Rights: Copyright © 2017 Mathematical Sciences Publishers

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