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Hessenberg Matrices and Integer Sequences Milan Janjić

We investigate a particular case of upper Hessenberg matrices, in which all subdiagonal elements are −1. Several mathematical objects may be represented as determinants of such matrices. We consider three type of matrices related to polynomials, generalized Fibonacci numbers, and a special kind of composition of natural numbers. Our objective is to find the combinatorial meaning of the coefficients of the characteristic polynomials. The coefficients of the characteristic polynomials of matrices of the first type, that is, the sums of principal minors, are related to some binomial identities. The characteristic polynomials of matrices of the second kind give, as a particular case, the so-called convolved Fibonacci numbers defined by Riordan [4]. Coefficients of the characteristic polynomial of matrices of the third kind are connected with a special kind of composition of natural numbers, in which there are two different types of ones. These were introduced by Deutsch [2], and studied by Grimaldi [3].