论文标题
$ Q，T $ -CATALAN数字通过本地连锁店的链条分解
Chain Decompositions of $q,t$-Catalan Numbers via Local Chains
论文作者
论文摘要
$ q，t $ -catalan编号$ \ mathrm {cat} _n（q，t）$枚举$ n \ times n $ triangle中的整数分区，其DINV和外部区域统计信息。论文[LLL18（Lee，Li，Loehr，Siam J.离散数学32（2018））提出了一种新方法来理解对称属性$ \ MATHRM {CAT} _n（q，q，q，t）= \ MATHRM {CAT} _n（t，Q）$，基于分配所有Integer chintitions infintitions infintitions infins infins chnitient chnections of infine ch。每个这样的全局链$ \ MATHCAL {C}_μ$都有相反的链$ \ Mathcal {C} _ {μ^*} $;这些合并以给出了$ \ mathrm {cat} _n（q，t）$的新小切片，$ q $和$ t $。在这里，我们通过开发一种新的通用方法来构建全局链$ \ Mathcal {C}_μ$从称为本地链的较小元素来推进[LLL18]的议程。我们为当地连锁店定义了一个局部相反的物业，这意味着全球连锁店所需的相反属性。与相应的全球属性相比，在特定情况下，该本地属性更容易验证。我们将此机械应用于最多$ 11 $的赤字分区构建所有全球链。这证明，对于所有$ n $，$ \ mathrm {cat} _n（q，t）$的条款至少$ \ binom {n} {2} {2} -11 $在$ q $和$ t $中对称。
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 [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 all integer partitions into infinite chains. Each such global chain $\mathcal{C}_μ$ has an opposite chain $\mathcal{C}_{μ^*}$; 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 [LLL18] by developing a new general method for building the global chains $\mathcal{C}_μ$ 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 $\binom{n}{2}-11$ are symmetric in $q$ and $t$.
