Cyclic Maya Diagrams and Rational Solutions of Higher Order Painlevé Systems

This paper focuses on the construction of rational solutions for the A_2n-Painlevé system, also called the Noumi-Yamada system, which are considered the higher order generalizations of PIV Painlevé. In this even case, we introduce a method to construct the rational solutions based on cyclic dressing chains of Schrödinger 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 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.