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

Diaz-Lopez, Alexander; Harris, Pamela E.; Insko, Erik; Perez-Lavin, Darleen

Involve 10, No. 2, 263-290 (2017).

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)

Keywords:

binomial coefficient; peak; permutation; signed permutation; permutation pattern

Citations:

Zbl 1322.60208