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arXiv · 1410.8408

On the cardinality of a factor set in the symmetric group

Krasimir Yordzhev
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Abstract

Let $n$ be a positive integer, $σ$ be an element of the symmetric group $\mathcal{S}_n$ and let $σ$ be a cycle of length $n$. The elements $α,β\in \mathcal{S}_n$ are $σ$-equivalent, if there are natural numbers $k$ and $l$, such that $σ^k α=βσ^l$, which is the same as the condition to exist natural numbers $k_1$ and $l_1$, such that $α= σ^{k_1} βσ^{l_1}$. In this work we examine some properties of the so defined equivalence relation. We build a finite oriented graph $Γ_n$ 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.

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Krasimir Yordzhev. 2014-10-30. On the cardinality of a factor set in the symmetric group. https://doi.org/10.1142/s1793557114500272

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