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Let G be a finite group. Let pi be a permutation from S{n}. We study thedistribution of probabilities of equality a{1} a{2}...a{n-1}a{n}=a{pi{1}}^{epsilon{1}}a{pi_{2}}^{epsilon{2}}...a{pi{n-1}}^{epsilon_{n-1}} a_{pi_{n}}^{epsilon{n}},when pi varies over all the permutations in S{n}, and epsilon{i} varies overthe set {+1, -1}. By the paper "Hultman Numbers and Generalized CommutingProbability in Finite Groups" (2017), The case where all epsilon{i} are +1 ledto a close connection to Hultman numbers. In this paper we generalize theresults, permitting epsilon{i} to be -1. We describe the spectrum of theprobabilities of signed permutation equalities in a finite group G. Thisspectrum turns out to be closely related to the partition of 2^{n}*n! into asum of the corresponding signed Hultman numbers.

Signed Hultman Numbers and Signed Generalized Commuting Probability in Finite Groups