Hultman Numbers and Generalized Commuting Probability in Finite Groups

Let $G$ be a finite group and $\pi$ be a permutation from $S_{n}$. We investigate the distribution of the probabilities of the equality $$a_{1}a_{2}\cdots a_{n-1}a_{n}=a_{\pi_{1}}a_{\pi_{2}}\cdots a_{\pi_{n-1}}a_{\pi_{n}}$$ when $\pi$ varies over all the permutations in $S_{n}$. The probability $$Pr_{\pi}(G)=Pr(a_{1}a_{2}\cdots a_{n-1}a_{n}=a_{\pi_{1}}a_{\pi_{2}}\cdots a_{\pi_{n-1}}a_{\pi_{n}})$$ is identical to $Pr_{1}^{\omega}(G)$, with $$\omega=a_{1}a_{2}...a_{n-1}a_{n}a_{\pi_{1}}^{-1}a_{\pi_{2}}^{-1}\cdots a_{\pi_{n-1}}^{-1}a_{\pi_{n}}^{-1},$$ as it is defined in \cite{DasNath1} and \cite{NathDash1}. The notion of commutativity degree, or the probability of a permutation equality $a_{1}a_{2}=a_{2}a_{1}$, for which $n=2$ and $\pi=\langle2\;\;1\rangle$, was introduced and assessed by P. Erd\"{o}s and P. Turan in \cite{ET} in 1968 and by W. H. Gustafson in \cite{G} in 1973. In \cite{G} Gustafson establishes a relation between the probability of $a_{1},a_{2}\in G$ 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 $\pi$, depends only on the number $c(Gr(\pi))$ of the alternating cycles in the cycle graph $Gr(\pi)$ of $\pi$. The cycle graph of a permutation was introduced by V. Bafna and P. A. Pevzner in \cite{BP}. We describe the spectrum of the probabilities of permutation equalities in a finite group as $\pi$ varies over all the elements of $S_{n}$. This spectrum turns-out to be closely related to the partition of $n!$ into a sum of the corresponding Hultman numbers.