我们说对称群$\mathfrak{S}_n$中的置换$π=π_1π_2\cdotsπ_n$如果$π_{i-1}<π_i>π_{i+1}$在索引$i$处有一个峰，我们令$P(π)=\{i \in \{1, 2, \ldots, n\} \, \vert \, \mbox{$ i $ is a peak of $ π $}\}$。给定一个正整数集合$S$，我们让$P (S; n)$表示由所有排列$π$组成的$\mathfrak{S}_n$的子集，其中$P(π) =S$。2013年，Billey、Burdzy和Sagan证明了$|P(S;n)| = p(n)2^{n-\lvert S\rvert-1}$，其中$p(n)$是次数为$\max(S)- 1$的多项式。在2014年，Castro-Velez等人认为$B_n$型的Coxeter群是$n$字母上的带符号置换群，并证明了$\lvert P_B(S;n)\rvert=p(n)2^{2n-|S|-1}$，其中$p(n)$是相同次数的$\max(S)-1$多项式。在本文中，我们分区的集合$P(S;n) \subset \mathfrak{S}_n$研究的比利，Burdzy，和Sagan到子集的$P(S;n)$的置换与峰值集$S$结束的上升到一个固定的整数$k$或下降，并提供多项式公式，这些子集的基数。将李型Coxeter群$C_n$和$D_n$嵌入到$\mathfrak{S}_{2n}$中后，我们将这些群划分成置换丛$π_1π_2 \cdotsπ_n|π_{n+1}\cdots π_{2n}$，使得$π_1π_2\cdots π_n$与某个置换$σ_1σ_2\cdotsσ_n \in \mathfrak{S}_n$具有相同的相对阶.这使我们能够通过将枚举简化为对称群中的计算以及帕斯卡三角形各行的总和，来计算具有给定峰值集$S$的$C_n$和$D_n$类型中的排列数。
英文摘要：
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.
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