Number of Compositions and Convolved Fibonacci numbers

TL;DR

This paper explores the relationship between Fibonacci numbers, convolved Fibonacci numbers, and compositions of natural numbers, providing new identities and explicit formulas through the analysis of Hessenberg matrices.

Contribution

It introduces new identities involving Fibonacci and convolved Fibonacci numbers and links compositions of natural numbers to Hessenberg matrices with Fibonacci determinants.

Findings

Determinants of specific Hessenberg matrices are Fibonacci numbers.

Sum of principal minors relates to convolved Fibonacci numbers.

Explicit formula for compositions with fixed number of ones in terms of Fibonacci convolutions.

Abstract

We consider two type of upper Hessenberg matrices which determinants are Fibonacci numbers. Calculating sums of principal minors of the fixed order of the first type leads us to convolved Fibonacci numbers. Some identities for these and for Fibonacci numbers are proved. We also show that numbers of compositions of a natural number with fixed number of ones appear as coefficients of characteristic polynomial of a Hessenberg matrix which determinant is a Fibonacci number. We derive the explicit formula for the number of such compositions, in terms of convolutions of Fibonacci numbers.