Some array polynomials over special monoid presentations

In a recent joint paper (Cevik et al. in Hacet. J. Math. Stat., acceptted), the authors have investigated the p-Cockcroft property (or, equivalently, efficiency) for a presentation, say P_E, of the semi-direct product of a free abelian monoid rank two by a finite cyclic monoid. Moreover, they have presented sufficient conditions on a special case for 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 P_E can actually be represented by characteristic polynomials. After that, some connections between representative characteristic polynomials and generating functions in terms of array polynomials over the presentation _PE will be pointed out. Through indicated connections, the existence of an equivalence among each generating function in itself is claimed studied in this paper.