K-Factorizations of the full cycle and generalized Mahonian statistics on k-forests

Abstract

We develop direct bijections between the set Fnk of minimal factorizations of the long cycle (01⋯kn) into (k+1)-cycle factors and the set Rnk of rooted labelled forests on vertices {1,…,n} with edges coloured with {0,1,…,k−1} that map natural statistics on the former to generalized Mahonian statistics on the latter. In particular, we examine the generalized major index on forests Rnk and show that it has a simple and natural interpretation in the context of factorizations. Our results extend those in [8], which treated the case k=1 through a different approach, and provide a bijective proof of the equidistribution observed by Yan [16] between displacement of k-parking functions and generalized inversions of k-forests.