Peaks Sets of Classical Coxeter Groups

TL;DR

This paper studies the enumeration of permutations with specific peak sets across classical Coxeter groups, providing polynomial formulas and combinatorial partitions that generalize earlier symmetric group results.

Contribution

It extends peak set enumeration to Coxeter groups of types B, C, and D, introducing polynomial formulas and combinatorial partitions for these groups.

Findings

Derived polynomial formulas for peak set counts in types B, C, and D.

Partitioned permutations based on ascent/descent to fixed integers.

Reduced enumeration in Coxeter groups to calculations in symmetric groups.

Abstract

We say a permutation $\pi=\pi_1\pi_2\cdots\pi_n$ in the symmetric group $\mathfrak{S}_n$ has a peak at index $i$ if $\pi_{i-1}<\pi_i>\pi_{i+1}$ and we let $P(\pi)=\{i \in \{1, 2, \ldots, n\} \, \vert \, \mbox{$i$ is a peak of $\pi$}\}$. Given a set $S$ of positive integers, we let $P (S; n)$ denote the subset of $\mathfrak{S}_n$ consisting of all permutations $\pi$, where $P(\pi) =S$. 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 $\pi_1\pi_2 \cdots\pi_n|\pi_{n+1}\cdots \pi_{2n}$ such that $\pi_1\pi_2\cdots \pi_n$ has the same relative order as some permutation $\sigma_1\sigma_2\cdots\sigma_n \in \mathfrak{S}_n$. 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.