## Peak sets of classical Coxeter groups.(English)Zbl 1350.05003

Summary: 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 $$P(\pi)=\{i \in \{1, 2,\dots, n\} \mid i \text{ is a peak of }\pi\}$$. 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 $$P(\pi) =S$$. S. Billey et al. [Ann. Appl. Probab. 25, No. 4, 1729–1779 (2015; Zbl 1322.60208)] proved $$|P(S;n)| = p(n)2^{n-|S|-1}$$, where $$p(n)$$ is a polynomial of degree $$\max(S){-}1$$. F. Castro-Velez et al. [“Number of permutations with same peak set for signed permutations”, Preprint, arXiv:1308.6621] 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 S. Billey et al. [loc. cit.] into subsets of permutations that end with an ascent to a fixed integer $$k$$ (or a descent to a fixed integer $$k$$) and provide polynomial formulas for the cardinalities of these subsets. After embedding the Coxeter groups of Lie types $$C_n$$ 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.

### MSC:

 05A05 Permutations, words, matrices 05A10 Factorials, binomial coefficients, combinatorial functions 05A15 Exact enumeration problems, generating functions 20F55 Reflection and Coxeter groups (group-theoretic aspects)

Zbl 1322.60208
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