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

Abstract

We develop a direct bijections between the set F k n of minimal factorizations of the long cycle (0 1 • • • kn) into (k + 1)-cycle factors and the set R k n 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 R k n and show that it has a simple and natural interpretation in the context of factorizations. Our results extend those in [IR21], which treated the case k = 1 through a different approach, and provide a bijective proof of the equidistribution observed by Yan [Yan97] between displacement of k-parking functions and generalized inversions of k-forests.