PAR-FUNCTIONS OF TOEPLITZ MATRICES AND PARTITION POLYNOMIALS

Abstract

This paper explores certain properties of the par and par^{+} functions of Toeplitz matrices. These functions are studied in tandem due to their many shared properties. Since the combinatorial foundation of these functions lies in ordered partitions of a natural number into non-negative integers, it becomes possible to represent them as partition polynomials and to construct recursive algorithms for their computation. In addition to a brief introduction to these functions, the paper presents a recurrence relation for computing the  par-functions of Toeplitz matrices, which enables the unification of a broad class of linear recurrence relations. As linear recurrence relations are often related to partition polynomials, the representation of these functions as partition polynomials is also studied. The article includes an example that utilizes the fact that the multilinear polynomials of par^+ and par-functions of square matrices contain 2^(n-1) terms, with the par-function comprising half positive and half negative terms. Two combinatorial identities are derived using a  Toeplitz matrix whose entries are all equal to one.