ABSTRACTGiven a permutation , we say an index i is a peak if πi − 1 πi + 1. Let P(π) denote the set of peaks of π. Given any set S of positive integers, define . Billey–Burdzy–Sagan showed that for all fixed subsets of positive integers S and sufficiently large n, for some polynomial pS(x) depending on S. They conjectured that the coefficients of pS(x) expanded in a binomial coefficient basis centered at max (S) are all positive. We show that this is a consequence of a stronger conjecture that bounds the modulus of the roots of pS(x). Furthermore, we give an efficient explicit formula for peak polynomials in the binomial basis centered at 0, which we use to identify many integer roots of peak polynomials along with certain inequalities and identities.
Another new result in @cite_7 shows that the number of permutations with the same peak set for signed permutations can be enumerated using the peak polynomial @math for unsigned permutations. We present an alternate proof that can be used to reduce many signed permutation statistic problems to unsigned permutation statistic problems. We denote the group of signed permutations as @math .
@cite_7
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A signed permutation = in the hyperoctahedral group B_n is a word such that each -n, , -1, 1, , n and | |, | |, , | | = 1,2, ,n . An index i is a peak of if i-1 i+1 and P_B( ) denotes the set of all peaks of . Given any set S, we define P_B(S,n) to be the set of signed permutations B_n with P_B( ) = 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 =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.