Minimal Factorizations of a Cycle and Central Multiplicative Functions on the Infinite Symmetric Group

We show that the number of factorizationsσ=χ1…χrof a cycle of lengthninto a product of cycles of lengthsa1, …, ar, with ∑rj=1 (aj−1)=n−1, is equal tonr−1. This generalizes a well known result of J. Denes, concerning factorizations into a product of transpositions. We investigate some consequences of this result, for central multiplicative functions on the infinite symmetric group, and use them to give a new proof of a recent result of A. Nica and R. Speicher on non-crossing partitions.