Determinants and Compositions of Natural Numbers

TL;DR

This paper explores the relationship between certain structured matrices and the enumeration of natural number compositions, providing formulas and connections to matrix characteristic polynomials.

Contribution

It introduces new formulas linking weak compositions with zeroes to compositions without zeroes and relates these to matrix coefficients.

Findings

Derived a formula expressing weak compositions with fixed zeroes in terms of compositions without zeroes

Established a relationship between weak compositions and characteristic polynomial coefficients

Provided explicit formulas for specific types of weak compositions

Abstract

We consider a particular type of matrices which belong at the same time to the class of Hessenberg and Toeplitz matrices, and whose determinants are equal to the number of a type of compositions of natural numbers. We prove a formula in which the number of weak compositions with a fixed number of zeroes is expressed in terms of the number of compositions without zeroes. Then we find a relationship between weak compositions and coefficients of characteristic polynomials of appropriate matrices. Finally, we prove three explicit formulas for weak compositions of a special kind.