Some array polynomials over special monoid presentations

In a recent joint paper \cite{CDCM}, the authors have been investigated the $p$-Cockcroft property (or, equivalently, efficiency) for a presentation, say ${\mathcal P}_E$, of the semi-direct product of a free abelian monoid rank two by a finite cyclic monoid. Moreover, they have been presented sufficient conditions on a special case for ${\mathcal P}_E$ to be minimal whilst it is inefficient. In this paper, by considering these results, we first show that the presentations of the form ${\mathcal P}_E$ can actually be represented by characteristic polynomials. After that, it will be pointed out some connections between representative characteristic polynomials and generating functions in terms of array polynomials over the presentation ${\mathcal P}_E$. Through indicated connections, it is claimed the existence of an equivalence among each generating functions in itself that studied in this paper.