Peaks Sets of Classical Coxeter 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 $P(π)=\{i \in \{1, 2, \ldots, n\} \, \vert \, \mbox{$i$isapeakof$π$}\}$. 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 $|P(S;n)|=p(n)2^{n-|S|-1}$, 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 $|P_B(S;n)|=p(n)2^{2n-|S|-1}$ 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.

24 pages

References in corpus (4)

• On meteors, earthworms and WIMPs
• Coefficients and roots of peak polynomials
• The most frequent peak set of a random permutation
• Peaks Sets of Classical Coxeter Groups

Cited by in corpus (1)

• Peaks Sets of Classical Coxeter Groups