  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.
