Determinants and Compositions of Natural Numbers

Details Determinants and Compositions of Natural Numbers Determinants and Compositions of Natural Numbers

2010-07-04

arxiv.org/abs/1007.0526v1

Mathematics

Combinatorics

Scientific paper

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.

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