On the construction of permutations of a given type kept fixed by conjugation (Q1803875)

Let \(\overline\alpha,\overline\beta\) be two partitions of \(n\), and \(\beta\in S_ n\) be a permutations of type \(\overline\beta\). The authors construct and enumerate the set \(\{\alpha\in S_ n\mid\alpha\) of type \(\overline\alpha\), \(\beta\alpha\beta^{-1}=\alpha\}\) by obtaining some recurrence formulas. The number of references is impressive.