Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: mtrl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-th
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The categories of this paper are: bayes-an
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The categories of this paper are: q-fin
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The categories of this paper are: atom-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: dg-ga
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The categories of this paper are: solv-int
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The categories of this paper are: math-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-bio