Hultman Numbers and Generalized Commuting Probability in Finite Groups

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Let G be a finite group and π be a permutation from Sₙ. We investigate the distribution of the probabilities of the equality a₁a₂ aₙ₋₁aₙ=aπ₁aπ₂ aπₙ₋₁aπₙ when π varies over all the permutations in Sₙ. The probability Prπ(G)=Pr(a₁a₂ aₙ₋₁aₙ=aπ₁aπ₂ aπₙ₋₁aπₙ) is identical to Pr₁ω(G), with ω=a₁a₂...aₙ₋₁aₙaπ₁⁻¹aπ₂⁻¹ aπₙ₋₁⁻¹aπₙ⁻¹, as it is defined in DasNath1 and NathDash1. The notion of commutativity degree, or the probability of a permutation equality a₁a₂=a₂a₁, for which n=2 and π=21, was introduced and assessed by P. Erdös and P. Turan in ET in 1968 and by W. H. Gustafson in G in 1973. In G Gustafson establishes 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 the probabilities of general permutation equalities in terms of these parameters. It turns out that this probability, for a permutation π, depends only on the number c(Gr(π)) of the alternating cycles in the cycle graph Gr(π) of π. The cycle graph of a permutation was introduced by V. Bafna and P. A. Pevzner in BP. We describe the spectrum of the probabilities of permutation equalities in a finite group as π varies over all the elements of Sₙ. This spectrum turns-out to be closely related to the partition of n! into a sum of the corresponding Hultman numbers.