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 π if 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 [4], 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)2 2n−|S|−1 where p(n) is the same polynomial found in [4] 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.

No abstract

No abstract