Abstract
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.
Citation
Alexander Diaz-Lopez. Pamela E. Harris. Erik Insko. Darleen Perez-Lavin. "Peak sets of classical Coxeter groups." Involve 10 (2) 263 - 290, 2017. https://doi.org/10.2140/involve.2017.10.263
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