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.


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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