Number of permutations with same peak set for signed permutations

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A signed permutation π= π₁π₂ πₙ in the hyperoctahedral group Bₙ is a word such that each πᵢ ∈ -n,, -1, 1,, n and |π₁|, |π₂|,, |πₙ| = 1,2,,n. An index i is a peak of πif πᵢ₋₁<πᵢ>πᵢ₊₁ and PB(π) denotes the set of all peaks of π. Given any set S, we define PB(S,n) to be the set of signed permutations π∈ Bₙ with PB(π) = S. In this paper we are interested in the cardinality of the set PB(S,n). In 2012, Billey, Burdzy and Sagan investigated the analogous problem for permutations in the symmetric group, Sₙ. In this paper we extend their results to the hyperoctahedral group; in particular we show that #PB(S,n) = p(n)2²ⁿ⁻|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 π₀=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.