Number of permutations with same peak set for signed permutations

Let Bn be the group of all signed permutations of [n]. A signed permutation has a peak in a position i=2,…,n−1 if πi−1<πi>πi+1. Let P(π) be the set of peaks of π, P(S,n) be the set of signed permutations π∈Bn such that P(π)=S, and #P(S,n) be the cardinality of P(S,n). We show #P(∅,n)=22n−1 and #P(S,n)=p(n)22n−|S|−1 where p(n) is some polynomial. We also consider the case in which we add a zero at the beginning of the permutation to also allow peaks at position i=1.

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