  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 $\mu$.
