Number of permutations with same peak set for signed permutations

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.