Signed Hultman numbers and signed generalized commuting probability in finite groups

Let G be a finite group. Let π be a permutation from S_n. We study the distribution of probabilities of equalitya1a2⋯an-1an=aπ1ϵ1aπ2ϵ2⋯aπn-1ϵn-1aπnϵn, when π varies over all the permutations in S_n, and ϵ_i varies over the set { + 1 , - 1 }. By [7], the case where all ϵ_i are + 1 led to a close connection to Hultman numbers. In this paper, we generalize the results, permitting ϵ_i to be - 1. We describe the spectrum of the probabilities of signed permutation equalities in a finite group G. This spectrum turns out to be closely related to the partition of 2 ⁿ· n! into a sum of the corresponding signed Hultman numbers.