On the cardinality of a factor set in the symmetric group

Let $n$ be a positive integer, $\sigma$ be an element of the symmetric group ${\mathcal{S}}_n$ and let $\sigma$ be a cycle of length $n$. The elements $\alpha ,\beta \in $\mathcal{S}$_n$are$\sigma$-equivalent,iftherearenaturalnumbers$k$and$l$,suchthat$\sigma^k \alpha =\beta\sigma^l$, which is the same as the condition to exist natural numbers $k_1$ and $l_1$, such that $\alpha = \sigma^{k_1} \beta\sigma^{l_1}$. In this work we examine some properties of the so defined equivalence relation. We build a finite oriented graph ${\mathrm{\Gamma }}_n$ with the help of which is described an algorithm for solving the combinatorial problem for finding the number of equivalence classes according to this relation.