Chain Decompositions of $q,t$-Catalan Numbers via Local Chains

TL;DR

This paper introduces a new method to construct global chains from local chains to analyze the symmetry of $q,t$-Catalan numbers, proving partial symmetry for high-degree terms.

Contribution

It develops a general approach to build global chains from local chains, simplifying the verification of symmetry properties in $q,t$-Catalan numbers.

Findings

Constructed all global chains for partitions with deficit ≤ 11

Proved symmetry in $q,t$-Catalan numbers for high-degree terms

Provided a new framework for analyzing combinatorial symmetries

Abstract

The $q,t$-Catalan number $\mathrm{Cat}_n(q,t)$ enumerates integer partitions contained in an $n\times n$ triangle by their dinv and external area statistics. The paper [LLL18 (Lee, Li, Loehr, SIAM J. Discrete Math. 32(2018))] proposed a new approach to understanding the symmetry property $\mathrm{Cat}_n(q,t)=\mathrm{Cat}_n(t,q)$ based on decomposing the set of all integer partitions into infinite chains. Each such global chain $\mathcal{C}_{\mu}$ has an opposite chain $\mathcal{C}_{\mu^*}$; these combine to give a new small slice of $\mathrm{Cat}_n(q,t)$ that is symmetric in $q$ and $t$. Here we advance the agenda of [LLL18] by developing a new general method for building the global chains $\mathcal{C}_{\mu}$ from smaller elements called local chains. We define a local opposite property for local chains that implies the needed opposite property of the global chains. This local property is much easier to verify in specific cases compared to the corresponding global property. We apply this machinery to construct all global chains for partitions with deficit at most $11$. This proves that for all $n$, the terms in $\mathrm{Cat}_n(q,t)$ of degree at least $\binom{n}{2}-11$ are symmetric in $q$ and $t$.