New combinatorial proof of Gaussian polynomial and the monotonicity of the Garvan's $k$-rank

Abstract

Gaussian polynomial, which is also known as $q$-binomial coefficient, is one of the fundamental concepts in the theory of partitions. Zeilberger provided a combinatorial proof of Gaussian polynomial, which is called Algorithm Z by Andrews and Bressoud. In this paper, we provide a new bijection on Gaussian polynomial, which leads to a refinement of Algorithm Z. Moreover, using this bijection, we provide an alternative proof of generalized Rogers-Ramanujan identity, which was first proved by Bressoud and Zeilberger. Furthermore, we give a combinatorial proof of the monotonicity property of Garvan's $k$-rank, which is a generalization of Dyson's rank and Andrews-Garvan's crank.