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

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 NU that builds the tail part of each chain $mathcal{C}_{mu}$. Our first main contribution here is to extend $NU$ and construct larger second-order tails for each chain. Second, we introduce new classes of partitions (flagpole partitions and generalized flagpole partitions) and give a recursive construction of the full chain $mathcal{C}_{mu}$ for generalized flagpole partitions $mu$.