Permutations as minimal powers of a single-cycle class-sum (Q1902969)

Let \(\mathbb{C} S_n\) be the class-algebra of the symmetric group \(S_n\). It is well-known that the structure constants of the class-algebra of any group \(G\) are completely determined by its table of characters [see \textit{G. James} and \textit{M. Liebeck}, Representations and Characters of Groups, Cambridge Univ. Press, Cambridge (1993; Zbl 0792.20006)]. Nevertheless, since tables of characters of symmetric groups are quite difficult to compute, the problem arise to find other ways to evaluate structure constants of \(\mathbb{C} S_n\). Note that when a permutation \(\pi_n\) of type \((1)^{\ell_1} (2)^{\ell_2} \dots (n)^{\ell_n}\) is multiplied by a \(p\)-cycle \((p)\) which possesses one index in common with each one of \(p\) cycles in \(\pi\), these \(p\) cycles merge into one. In the paper the corresponding structure constant in the product of classes \([(p)]_n \cdot [\pi]\) is derived. This result is used to evaluate the structure constant in \([(p)]^k_n\), corresponding to a class with the smallest possible number of cycles. This is a generalization of the case \(p = 2\) that was studied in 1959 by Dénes [see also \textit{P. Moszkowski}, Eur. J. Comb. 10, No. 1, 13-16 (1989; Zbl 0672.05022) and \textit{I. P. Goulden} and \textit{S. Pepper}, Discrete Math. 113, No. 1-3, 263-268 (1993; Zbl 0779.05017)].