Chain decompositions of q,t-Catalan numbers: tail extensions and flagpole partitions

TL;DR

This paper advances the combinatorial understanding of q,t-Catalan numbers by developing a chain decomposition framework for partitions, introducing flagpole partitions, and providing recursive constructions and structural insights.

Contribution

It extends the chain tail construction, introduces flagpole partitions, and offers recursive methods for chain building in the study of q,t-Catalan numbers.

Findings

Extended tail construction for chains

Introduction of flagpole and generalized flagpole partitions

Recursive chain-building methods for specific partitions

Abstract

This article is part of an ongoing investigation of the combinatorics of $q,t$-Catalan numbers $\textrm{Cat}_n(q,t)$. We develop a structure theory for integer partitions based on the partition statistics dinv, deficit, and minimum triangle height. Our goal is to decompose the infinite set of partitions of deficit $k$ into a disjoint union of chains $\mathcal{C}_{\mu}$ indexed by partitions of size $k$. Among other structural properties, these chains can be paired to give refinements of the famous symmetry property $\textrm{Cat}_n(q,t)=\textrm{Cat}_n(t,q)$. Previously, we introduced a map that builds the tail part of each chain $\mathcal{C}_{\mu}$. Our first main contribution here is to extend this map to construct larger second-order tails for each chain. Second, we introduce new classes of partitions called flagpole partitions and generalized flagpole partitions. Third, we describe a recursive construction for building the chain $\mathcal{C}_{\mu}$ for a (generalized) flagpole partition $\mu$, assuming that the chains indexed by certain specific smaller partitions (depending on $\mu$) are already known. We also give some enumerative and asymptotic results for flagpole partitions and their generalized versions.