Peaks Sets of Classical Coxeter Groups

Abstract

We say a permutation $π=π_1π_2\cdotsπ_n$ in the symmetric group $\mathfrak{S}_n$ has a peak at index $i$ if $π_{i-1}<π_i>π_{i+1}$ and we let $P(π)=\{i \in \{1, 2, \ldots, n\} \, \vert \, \mbox{$i$ is a peak of $π$}\}$. Given a set $S$ of positive integers, we let $P (S; n)$ denote the subset of $\mathfrak{S}_n$ consisting of all permutations $π$, where $P(π) =S$. In 2013, Billey, Burdzy, and Sagan proved $|P(S;n)| = p(n)2^{n-\lvert S\rvert-1}$, 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 $\lvert P_B(S;n)\rvert=p(n)2^{2n-|S|-1}$ where $p(n)$ is the same polynomial of degree $\max(S)-1$. In this paper we partition the sets $P(S;n) \subset \mathfrak{S}_n$ studied by Billey, Burdzy, and Sagan into subsets of $P(S;n)$ of permutations with peak set $S$ that end with an ascent to a fixed integer $k$ or a descent and provide polynomial formulas for the cardinalities of these subsets. After embedding the Coxeter groups of Lie type $C_n$ and $D_n$ into $\mathfrak{S}_{2n}$, we partition these groups into bundles of permutations $π_1π_2 \cdotsπ_n|π_{n+1}\cdots π_{2n}$ such that $π_1π_2\cdots π_n$ has the same relative order as some permutation $σ_1σ_2\cdotsσ_n \in \mathfrak{S}_n$. 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.