Sur les représentations des permutations impaires. (On the representations of odd permutations) (Q1197622)

A simple combinatorial proof is given of the fact that all odd permutations on \(n\) letters can be represented in \(2(n-2)!\) different ways as a product of a cycle of length \(n\) and a cycle of length \(n-1\). The method of proof provides an algorithm for explicitly constructing all such representations. One can also use the method to construct representations of even permutations on \(n\) letters as a product of two cycles of length \(n\), cf. \textit{D. H. Husemoller} [Ramified coverings of Riemann surfaces, Duke Math. J. 29, 167-174 (1962; Zbl 0196.340)].