Sorting Index and Mahonian-Stirling Pairs for Labeled Forests

TL;DR

This paper extends the understanding of permutation-like statistics to labeled forests, introducing new measures like sorting index and cycle minima, and demonstrates their equidistribution with classical statistics, generalizing known permutation results.

Contribution

It introduces new statistics for labeled forests and signed forests, extending permutation results to these structures and establishing their equidistribution properties.

Findings

Pairs (inv, Bt-max), (sor, Cyc), and (maj, Cbt-max) are equidistributed.

Results generalize permutation statistics to labeled and signed forests.

New statistics like sorting index and cycle minima are introduced and studied.

Abstract

Bj\"orner and Wachs defined a major index for labeled plane forests and showed that it has the same distribution as the number of inversions. We define and study the distributions of a few other natural statistics on labeled forests. Specifically, we introduce the notions of bottom-to-top maxima, cyclic bottom-to-top maxima, sorting index, and cycle minima. Then we show that the pairs (inv, Bt-max), (sor, Cyc), and (maj, Cbt-max) are equidistributed. Our results extend the result of Bj\"orner and Wachs and generalize results for permutations. We also introduce analogous statistics for signed labeled forests and show equidistribution results which generalize results for signed permutations.