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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 welet $P(\pi)=\{i \in \{1, 2, \ldots, n\} \, \vert \, \mbox{$i$ is a peak of$\pi$}\}$. Given a set $S$ of positive integers, we let $P (S; n)$ denote thesubset of $\mathfrak{S}_n$ consisting of all permutations $\pi$, where $P(\pi)=S$. In 2013, Billey, Burdzy, and Sagan proved $|P(S;n)| = p(n)2^{n-\lvertS\rvert-1}$, 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 ofsigned permutations on $n$ letters and showed that $\lvertP_B(S;n)\rvert=p(n)2^{2n-|S|-1}$ 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 thesesubsets. 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 countthe 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 sumsacross the rows of Pascal's triangle.

Peaks Sets of Classical Coxeter Groups