Hultman numbers and generalized commuting probability in finite groups

Let G be a finite group and π be a permutation from Sn. We investigate the distribution of the probabilities of the equality a₁a₂⋯a_n−1a_n=a_π1a_π2⋯a_πn−1a_πn when π varies over all the permutations in Sn. The probability Pr_π(G)=Pr(a₁a₂⋯a_n−1a_n=a_π1a_π2⋯a_πn−1a_πn) is identical to Pr^ω ₁(G), with ω=a₁a₂…a_n−1a_n-1a_na⁻¹_πna⁻_πn-1… a⁻_π2a⁻_π1, which was defined and studied by Das and Nath. The notion of commutativity degree, or the probability of a permutation equality a₁a₂ = a₂a₁, for which n = 2 and π = ⟨2 1⟩ was introduced and assessed by Erdös and Turan in 1968 and by Gustafson in 1973. Gustafson established a relation between the probability of a1, a2 ∈ G commuting and the number of conjugacy classes in G. In this work we define several other parameters, which depend only on a certain interplay between the conjugacy classes of G, and compute probabilities of permutation equalities in terms of these parameters. It turns out that for a permutation π, the probability of its permutation equality depends only on the number c(Gr(π)) of alternating cycles in the cycle graph Gr(π) of π. The cycle graph of a permutation was introduced by Bafna and Pevzner, and the number of alternating cycles in it was introduced by Hultman. Hultman numbers are the numbers of different permutations with the same number of alternating cycles in their cycle graphs. We show that the spectrum of probabilities of permutation equalities in a generic finite group, as π varies over all the permutations in S_n, corresponds to partitioning n! as the sum of the corresponding Hultman numbers.