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The $q,t$-Catalan number $\mathrm{Cat}_n(q,t)$ enumerates integer partitionscontained in an $n\times n$ triangle by their dinv and external areastatistics. The paper [LLL18 (Lee, Li, Loehr, SIAM J. Discrete Math. 32(2018))]proposed a new approach to understanding the symmetry property$\mathrm{Cat}_n(q,t)=\mathrm{Cat}_n(t,q)$ based on decomposing the set of allinteger partitions into infinite chains. Each such global chain$\mathcal{C}_{\mu}$ has an opposite chain $\mathcal{C}_{\mu^*}$; these combineto give a new small slice of $\mathrm{Cat}_n(q,t)$ that is symmetric in $q$ and$t$. Here we advance the agenda of [LLL18] by developing a new general methodfor building the global chains $\mathcal{C}_{\mu}$ from smaller elements calledlocal chains. We define a local opposite property for local chains that impliesthe needed opposite property of the global chains. This local property is mucheasier to verify in specific cases compared to the corresponding globalproperty. We apply this machinery to construct all global chains for partitionswith deficit at most $11$. This proves that for all $n$, the terms in$\mathrm{Cat}_n(q,t)$ of degree at least $\binom{n}{2}-11$ are symmetric in $q$and $t$.

Chain Decompositions of $q,t$-Catalan Numbers via Local Chains