This paper focuses on the construction of rational solutions for the $A_{2n}$ Painlev\'e system, also called the Noumi-Yamada system, which are considered the higher order generalizations of PIV. In this even case, we introduce a method to construct the rational solutions based on cyclic dressing chains of Schr\"{o}dinger operators with potentials in the class of rational extensions of the harmonic oscillator. Each potential in the chain can be indexed by a single Maya diagram and expressed in terms of a Wronskian determinant whose entries are Hermite polynomials. We introduce the notion of cyclic Maya diagrams and we characterize them for any possible period, using the concepts of genus and interlacing. The resulting classes of solutions can be expressed in terms of special polynomials that generalize the families of generalized Hermite, generalized Okamoto and Umemura polynomials, showing that they are particular cases of a larger family.

中文翻译：

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高阶 Painlevé 系统的循环 Maya 图和合理解

本文重点讨论了$A_{2n}$Painlev\'e 系统（也称为 Noumi-Yamada 系统）的有理解的构建，该系统被认为是 PIV 的高阶泛化。在这种偶数情况下，我们引入了一种基于 Schr\"{o}dinger 算子的循环修整链构造有理解的方法，其势在谐振子的有理拓类中。链中的每个势可以由下式表示单个 Maya 图并用 Wronskian 行列式表示，其条目是 Hermite 多项式。我们引入了循环 Maya 图的概念，并使用属 (genus) 和隔行 (interlacing) 的概念来表征它们在任何可能的周期内。所得的解的类别可以是用特殊多项式表示，概括了广义 Hermite 的家族，