Odd permutations are nicer than even ones

International audienceWe give simple combinatorial proofs of some formulas for the number of factorizations of permutations in S n as a product of two n-cycles, or of an n-cycle and an (n−1)-cycle. ... The parameter number of cycles plays a central role in the algebraic theory of the symmetric group, however there are very few results giving a relationship between the number of cycles of two permutations and that of their product. ... The first results on the subject go back to Ore, Bertram, Stanley (see [13], [1] and [15]), who proved some existence theorems. These results allowed to obtain ..