On a Theorem of J. Shallit Concerning Fibonacci Partitions

TL;DR

This paper proves a determinant property of special matrices that, combined with previous work on Fibonacci partitions, strengthens existing results about the structure and enumeration of Fibonacci partitions.

Contribution

It establishes a new determinant result for specific tridiagonal matrices, enhancing the understanding of Fibonacci partitions and extending Shallit's earlier findings.

Findings

Proved a determinant property of special tridiagonal matrices.

Strengthened existing results on Fibonacci partitions.

Connected matrix determinants to Fibonacci partition enumeration.

Abstract

In this note I prove a~claim on determinants of some special tridiagonal matrices. Together with my result about Fibonacci partitions (arXiv:math/0307150), this claim allows one to prove one (slightly strengthened) Shallit's result about such partitions.