We say a permutation in the symmetric group has a peak at index if and we let . Given a set of positive integers, we let denote the subset of consisting of all permutations where . In 2013, Billey, Burdzy, and Sagan proved , where is a polynomial of degree . In 2014, Castro-Velez et al. considered the Coxeter group of type as the group of signed permutations on letters and showed that , where is the same polynomial of degree . In this paper we partition the sets studied by Billey, Burdzy, and Sagan into subsets of permutations that end with an ascent to a fixed integer (or a descent to a fixed integer ) and provide polynomial formulas for the cardinalities of these subsets. After embedding the Coxeter groups of Lie types and into , we partition these groups into bundles of permutations such that has the same relative order as some permutation . This allows us to count the number of permutations in types and with a given peak set by reducing the enumeration to calculations in the symmetric group and sums across the rows of Pascal’s triangle.
"Peak sets of classical Coxeter groups." Involve 10 (2) 263 - 290, 2017. https://doi.org/10.2140/involve.2017.10.263