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

TL;DR

This paper establishes bijections between minimal factorizations of a long cycle and colored rooted forests, linking natural statistics to generalized Mahonian statistics, extending previous work and providing combinatorial proofs of known distributions.

Contribution

It introduces new bijections connecting cycle factorizations and colored forests, generalizing earlier results for the case k=1, and interprets Mahonian statistics in this context.

Findings

Bijections between factorizations and forests are constructed.

Generalized major index is interpreted within factorizations.

Provides a combinatorial proof of distribution equivalences.

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.