Signed Hultman numbers and generalized commuting probability in finite groups

Let $G$ be a finite group, and let $\pi$ be a permutation from $S_n$. We study the distribution of probabilities of the equality

\[ a_1 a_2 \cdots a_n = a_{\pi_1}^{\epsilon_1} a_{\pi_2}^{\epsilon_2} \cdots a_{\pi_n}^{\epsilon_n}, \]

where $\pi$ varies over all the permutations in $S_n$, and $\epsilon_i$ varies either over the set $\{1, -1\}$ or over the set $\{1, b\}$, where $b$ is an involution in $G$ (two different cases of equations). The equation can also be written as

\[ a_1 a_2 \cdots a_n = a_{\pi_1} a_{\pi_2} \cdots a_{\pi_n}, \]

where $\pi$ is a signed permutation from $B_n$, and $a_{-i}$ is interpreted either as the inverse $a_i^{-1}$, in case $\epsilon_i \in \{1, -1\}$, or as the conjugate $b a_i b$, in case $\epsilon_i \in \{1, b\}$. The probability in the second case depends on the conjugacy class of the involution $b$.

First we consider the case in which all $\epsilon_i$ are $1$. It turns out that the probability, for a permutation $\pi$, depends only on the number of alternating cycles in the cycle graph of $\pi$, introduced by Bafna and Pevzner in 1998. We describe the spectrum of probabilities of permutation equalities in a finite group as $\pi$ varies over all permutations in $S_n$. We then generalize, letting a signed permutation vary over all the signed permutations in $B_n$, under the two interpretations outlined above. The spectrum turns out to be closely related to the partition of the number $2^{n}\cdot n!$ into a sum of the corresponding signed Hultman numbers (defined by Grusea and Labbare) when $\epsilon_i \in \{1, -1\}$, or into edge-signed Hultman numbers (introduced in this talk) when $\epsilon_i \in \{1, b\}$.