Chain Decompositions of q, t-Catalan Numbers: Tail Extensions and Flagpole Partitions

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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.