Rationality of generating functions of rook polynomials and permanents of Kronecker products of Toeplitz matrices and circulants with the matrix $J_k$ and their evaluation. I

TL;DR

This paper generalizes the theory of rook polynomials and permanents for circulant and Toeplitz matrices, focusing on their Kronecker products with the matrix J_k, and explores their generating functions and evaluations.

Contribution

The paper introduces a new generalized framework for rook polynomials and permanents of specific structured matrices and their Kronecker products, extending previous theories.

Findings

Derived explicit formulas for rook polynomials and permanents of the matrices considered.

Established properties of the generating functions associated with these polynomials.

Provided evaluation techniques for the generalized rook polynomials and permanents.

Abstract

In this paper we give a generalization created by the author of the theory of rook polynomials and permanents of circulants, Toeplits matrices and their submatrices.