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

2021/10/26 by John Irving, Irving, John, Amarpreet Rattan +1 · 1 citation
Mathematics · #05A05 #05A19 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2110.13906

openalex publication_date 2021/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We develop direct bijections between the set F_n^k of minimal factorizations of the long cycle (0 1 ⋯ kn) into (k+1)-cycle factors and the set R_n^k 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_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.

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