The q, t-Catalan number \({{\,\mathrm{Cat}\,}}_n(q,t)\) enumerates integer partitions contained in an \(n\times n\) triangle by their dinv and external area statistics. The paper by Lee et al. (SIAM J Discr Math 32:191–232, 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 all integer partitions into infinite chains. Each such global chain \(\mathcal {C}_{\mu }\) has an opposite chain \(\mathcal {C}_{\mu ^*}\); these combine to give a new small slice of \({{\,\mathrm{Cat}\,}}_n(q,t)\) that is symmetric in q and t. Here, we advance the agenda of Lee et al. (SIAM J Discr Math 32:191–232, 2018) by developing a new general method for building the global chains \(\mathcal {C}_{\mu }\) from smaller elements called local chains. We define a local opposite property for local chains that implies the needed opposite property of the global chains. This local property is much easier to verify in specific cases compared to the corresponding global property. We apply this machinery to construct all global chains for partitions with deficit at most \(11\). This proves that for all n, the terms in \({{\,\mathrm{Cat}\,}}_n(q,t)\) of degree at least \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) -11\) are symmetric in q and t.

的q， 吨-Catalan数\（{{\，\ mathrm {猫} \，}} _ N（Q，T）\）枚举整数分区包含在\（N \ n次\）三角形可以通过DINV和外部区域统计。Lee等人的论文。（SIAM J Discr Math 32：191–232，2018）提出了一种新的方法来理解对称属性\（{{\，\ mathrm {Cat} \，}} _ n（q，t）= {{\，\ mathrm {Cat} \，}} _ n（t，q）\）基于将所有整数分区的集合分解为无限链。每个这样的全局链\（\ mathcal {C} _ {\ mu} \）都有相反的链\（\ mathcal {C} _ {\ mu ^ *} \）；这些组合给出的一个新的小片\（{{\，\ mathrm {猫} \，}} _ N（Q，T）\）是在对称q和t。在这里，我们推进Lee等人的议程。（SIAM J Discr Math 32：191–232，2018），这是通过开发一种新的通用方法来从称为局部链的较小元素构建全局链\（\ mathcal {C} _ {\ mu} \）。我们为局部链定义了局部对立属性，这意味着需要全局链的对立属性。与相应的全局属性相比，在特定情况下，此本地属性要容易得多。我们应用这种机制来构造所有全局链，以用于最多具有\（11 \）的分区。这证明对于所有n，度数\（{{\，\ mathrm {Cat} \，}} _ n（q，t）\）中的项至少\（\ left（{\ begin {array} {c} n \\ 2 \ end {array}} \ right）-11 \）在q和t中是对称的。