The $(m,n)$-rational $q, t$-Catalan polynomials for $m=3$ and their $q,t$-symmetry

TL;DR

This paper introduces a new combinatorial statistic called skip on rational (3,n)-Dyck paths, provides an explicit formula for the associated rational q,t-Catalan polynomials when m=3, and proves their q,t-symmetry through a bijection that exchanges area and dinv.

Contribution

The paper defines the skip statistic, constructs an explicit formula for (3,n)-rational q,t-Catalan polynomials, and proves their q,t-symmetry using a novel bijection.

Findings

Defined the skip statistic on rational (3,n)-Dyck paths.

Provided an explicit formula for the (3,n)-rational q,t-Catalan polynomials.

Proved the q,t-symmetry of these polynomials for m=3.

Abstract

We introduce a new statistic, skip, on rational $(3,n)$-Dyck paths and define a marked rank word for each path when $n$ is not a multiple of 3. If a triple of valid statistics (area,skip,dinv) are given, we have an algorithm to construct the marked rank word corresponding to the triple. By considering all valid triples we give an explicit formula for the $(m,n)$-rational $q,t$-Catalan polynomials when $m=3$. Then there is a natural bijection on the triples of statistics (area,skips,dinv) which exchanges the statistics area and dinv while fixing the skip. Thus we prove the $q,t$-symmetry of $(m,n)$-rational $q, t$-Catalan polynomials for $m=3$.