Number of permutations with same peak set for signed permutations

TL;DR

This paper extends previous work on counting permutations with a given peak set from the symmetric group to signed permutations in the hyperoctahedral group, providing explicit formulas and polynomial characterizations.

Contribution

It generalizes the peak set enumeration from permutations to signed permutations, deriving a closed-form formula involving a polynomial and powers of two.

Findings

Derived a formula for the number of signed permutations with a given peak set.

Extended results to include permutations with a peak at the first position.

Provided explicit computations for special cases of the polynomial p(n).

Abstract

A signed permutation \pi = \pi_1\pi_2 \ldots \pi_n in the hyperoctahedral group B_n is a word such that each \pi_i \in {-n, \ldots, -1, 1, \ldots, n} and {|\pi_1|, |\pi_2|, \ldots, |\pi_n|} = {1,2,\ldots,n}. An index i is a peak of \pi if \pi_{i-1}<\pi_i>\pi_{i+1} and P_B(\pi) denotes the set of all peaks of \pi. Given any set S, we define P_B(S,n) to be the set of signed permutations \pi \in B_n with P_B(\pi) = S. In this paper we are interested in the cardinality of the set P_B(S,n). In 2012, Billey, Burdzy and Sagan investigated the analogous problem for permutations in the symmetric group, S_n. In this paper we extend their results to the hyperoctahedral group; in particular we show that #P_B(S,n) = p(n)2^{2n-|S|-1} where p(n) is the same polynomial found in by Billey, Burdzy and Sagan which leads to the explicit computation of interesting special cases of the polynomial p(n). In addition we have extended these results to the case where we add \pi_0=0 at the beginning of the permutations, which gives rise to the possibility of a peak at position 1, for both the symmetric and the hyperoctahedral groups.