Peak sets of classical Coxeter groups

Abstract

We say a permutation π = π1π2 · · · πn in the symmetric group Sn has a peak at index i if πi−1 < πi > πi+1 and we let P (π) = {i ∈ {1, 2, . . . , n} | i is a peak of π}. Given a set S of positive integers, we let P (S; n) denote the subset of Sn consisting of all permutations π, where P (π) = S. In 2013, Billey, Burdzy, and Sagan proved |P (S; n)| = p(n)2 n−|S|−1 , where p(n) is a polynomial of degree max(S) − 1. In 2014, CastroVelez et al. considered the Coxeter group of type Bn as the group of signed permutations on n letters and showed that |PB(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) ⊂ Sn 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 Cn and Dn into S2n, we partition these groups into bundles of permutations π1π2 · · · πn|πn+1 · · · π2n such that π1π2 · · · πn has the same relative order as some permutation σ1σ2 · · · σn ∈ Sn. This allows us to count the number of permutations in types Cn and Dn 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.

A proof of the peak polynomial positivity conjecture

“…One of their foundational results established that for an n-admissible set S |P S (n)| = p S (n)2 n−|S|−1 (1) where p S (x) is a polynomial depending on S, which they called the peak polynomial of S. It was shown that p S (x) has degree max(S) − 1, and that p S (x) takes on integral values when evaluated at integers [3, Theorem 1.1]. Similar observations were made for peak polynomials in other classical Coxeter groups (see the work of Castro-Velez, Diaz-Lopez, Orellana, Pastrana [9] and Diaz-Lopez, Harris, Insko, and Perez-Lavin [10]). Using the method of finite differences, Billey, Burdzy, and Sagan gave closed formulas for the peak polynomials p S (x) in various special cases.…”

Pinnacle sets of signed permutations

“…Shortly thereafter, their results were extended to permutations in type B by Castro-Velez et al [2] where it was shown that the number of permutations with a given peak set I is 2 2n−|I|−1 p(n), with p(n) the same as in [1] above. The second author and various collaborators went further by extending these results to types C and D [7], using peaks to study properties of the descent polynomial [6], and then initiating the study of peaks in the context of graphs [4].…”

A formula for enumerating permutations with a fixed pinnacle set

“…In this article, we study the Hamming metric, which measures the number of indices at which two permutations differ and the ℓ ∞ -metric, which measures the maximum componentwise difference between two permutations. Motivated by a paper that studies these metrics on sets of permutations with a given peak set [5], the main purpose of this article is to find the maximum Hamming and ℓ ∞ -metric when restricting to permutations that share a descent set-subsets of the form D(S; n) ⊂ S n for S ⊂ [n − 1]. Theorem 3.2 provides a complete characterization of the maximum Hamming metric on all sets D(S; n) and Theorems 4.1 and 4.2 provide the maximum ℓ ∞ -metric on D(S; n) for particular cases of S. Finding the maximum ℓ ∞ -metric on most sets D(S; n) is still an open problem.…”