Abordagens algébrica e combinatória para o polinômio de Gauss

Abstract

The main focus of this work is to explore the concept of the q-binomial coefficient together with its properties, approaching the algebraic and combinatorial aspects of the Gaussian polynomial emphasizing that this polynomial evaluated in the indeterminate q = 1, reduces to the binomial coefficient and in this case we say That this polynomial is an extension of the binomial coefficient. Introducing concepts and definitions for the construction of the Gaussian Polynomial, we highlight the elementary symmetric polynomials that have a close relationship with Girard's and Newton's relations. The combinatorial approach will be done by means of partitions at most m parts with each part smaller than or equal to N. Through this idea we will relate this concept with reticulated paths and tiling.