  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.

Combinatorics