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

TL;DR

The paper develops a refined combinatorial framework around Gaussian polynomials and partition statistics by constructing explicit bijections that interlink structured partition sets. Central to the work are the bijections $\psi$ and $\phi_M$, which enable refined Algorithm Z approaches and enable alternative proofs of classical identities, including a generalized Rogers–Ramanujan identity. A key outcome is a new combinatorial proof of the monotonicity of Garvan's $k$-rank, extending the Dysons–Atkin–Ono framework. Together, these results deepen the bijective understanding of $q$-binomial coefficients, partition statistics, and their connections to representation theory and modular-type identities.

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.

Figures (6)

• Figure 4.1: Decompose $\lambda$ into $3k$ partitions.
• Figure 4.2: Illustration of $2m$-Durfee rectangle of $\lambda$.
• Figure 4.3: Step 8 to construct $\overline{\alpha}$.
• Figure 5.1: Illustration of successive Durfee squares of $\pi$.
• Figure 5.2: Example of $k-1$ successive Durfee Squares.

Theorems & Definitions (66)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Corollary 1.9
• Theorem 1.10
• Proposition 2.4