LAUSR

Let G be a finite group. Let $$\pi $$ π be a permutation from $$S_{n}$$ S n . We study the distribution of probabilities of equality $$ a_{1}a_{2}\cdots a_{n-1}a_{n}=a_{\pi _{1}}^{\epsilon _{1} }a_{\pi _{2}}^{\epsilon _{2}}\cdots a_{\pi _{n-1}}^{\epsilon _{n-1}}a_{\pi _{n} }^{\epsilon _{n}},$$ a 1 a 2 ⋯ a n - 1 a n = a π 1 ϵ 1 a π 2 ϵ 2 ⋯ a π n - 1 ϵ n - 1 a π n ϵ n , when $$\pi $$ π varies over all the permutations in $$S_{n}$$ S n , and $$\epsilon _{i}$$ ϵ i varies over the set $$\{+1, -1\}$$ { + 1 , - 1 } . By [ 7 ], the case where all $$\epsilon _{i}$$ ϵ i are $$+1$$ + 1 led to a close connection to Hultman numbers. In this paper, we generalize the results, permitting $$\epsilon _{i}$$ ϵ i to be $$-1$$ - 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 $$2^{n}\cdot n!$$ 2 n · n ! into a sum of the corresponding signed Hultman numbers.