Major Index on Catalan Combinatorics

TL;DR

This paper investigates the distribution of inversion number and major index statistics across various Catalan combinatorial objects, establishing their equidistribution with corresponding statistics on Dyck paths.

Contribution

It demonstrates the equidistribution of these statistics on multiple Catalan objects, extending understanding of their combinatorial properties.

Findings

Inversion number and major index are equidistributed on Catalan objects.

The results unify various combinatorial structures under common statistical distributions.

Provides new insights into the symmetry of Catalan combinatorics.

Abstract

We study the two statistics, the inversion number and the major index, on Catalan combinatorial objects such as $r$-Dyck paths, $r$-Stirling permutations, non-crossing partitions, Dyck tilings, and symmetric Dyck paths. We show that they are equidistributed with the inversion number or the major index on Dyck paths.