Bivariate local permutation polynomials, their companions, and related enumeration results

We construct a new family of permutation group polynomials over finite fields of arbitrary characteristic, which are special types of bivariate local permutation polynomials. For this family, we explicitly construct their companion. We also determine the total number of permutation group polynomials of this form. Moreover, we resolve the problem of enumerating $e$-Klenian polynomials over finite fields for $e\geq 1$, a problem previously noted as nontrivial by Gutierrez and Urroz (2023). In addition, we provide the exact number of permutation group polynomials equivalent to our proposed permutation group polynomials, as well as the exact number of those permutation group polynomials equivalent to $e$-Klenian polynomials.