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## FRI-G27 PEAK SETS OF COXETER GROUPS OF CLASSICAL LIE TYPES

- Darleen Perez-Lavin ;
- Erik Insko ;
- Pamela Harris ;
- Alex Diaz ;

FRI-G27

PEAK SETS OF COXETER GROUPS OF CLASSICAL LIE TYPES

__Darleen Perez-Lavin__^{1}, Erik Insko^{1}, Pamela Harris^{2}, Alex Diaz^{3}.

^{1}*Florida Gulf Coast University, Fort Myers, FL*, * ^{2}United States Military Academy, West Point, NY*,

*.*

^{3}University of^{ }Notre Dame, Notre Dame, INWe say a permutation π = π1 π 2 ... πn in the symmetric group Sn has a peak at index i if

π{i-1} < πi > π{i+1} and we let P(π) = {π in [n] | {i is a peak of π}}. Given a set S of positive integers, we let P(Sn) denote the subset of Sn consisting of all permutations π, where P(π) = S. In 2013, Billey, Burdzy, and Sagan proved |P(Sn)| = 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(Sn)]|=p(n)2{2n-|S|-1}. We partitioned the set P(Sn) ⊂ Sn studied by Billey, Burdzy, and Sagan into subsets P[(Sn), a(k)] ⊂ Sn of permutations with peak set S that ends with an ascent to a fixed integer k and provided polynomial formulas for the cardinalities of these subsets. After embedding the Coxeter groups of Lie type Cn and D;n into S{2n}, we partitioned these groups into bundles of permutations π1... πn π{n+1}... π{2n} that have the same relative order as some permutation σ1 σ2 ... σn in Sn. This allowed 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 type A and sums across the rows of Pascal's triangle.

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