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

Abstract

We develop direct bijections between the set $F_n^k$ of minimal factorizations of the long cycle $(0\,1\,\cdots\, kn)$ into $(k+1)$-cycle factors and the set $R_n^k$ of rooted labelled forests on vertices $\{1,\ldots,n\}$ with edges coloured with $\{0,1,\ldots,k-1\}$ that map natural statistics on the former to generalized Mahonian statistics on the latter. In particular, we examine the generalized major index on forests $R_n^k$ and show that it has a simple and natural interpretation in the context of factorizations. Our results extend those by the present authors (2021), which treated the case $k=1$ through a different approach, and provide a bijective proof of the equidistribution observed by Yan (1997) between displacement of $k$-parking functions and generalized inversions of $k$-forests.