Combinatorial proof of a result on generalized overcubic partitions and related conjectures

TL;DR

This work addresses congruences for generalized overcubic partitions bar a_c(n), extending prior results and confirming select conjectures. It provides a purely combinatorial proof of a modulo-4 congruence (Theorem p3thm1) by partitioning overcubic partitions into single-part and multi-part classes and analyzing divisor-based counts, with a case analysis on n being a square, twice a square, or neither. The paper also offers elementary proofs for several modulo-3 conjectures using Ramanujan theta-function identities and 3-dissections, connecting these results to the modulo-4 framework. Together, the results deepen understanding of arithmetic properties of generalized overcubic partitions and demonstrate effective combinatorial techniques alongside theta-function methods.

Abstract

Extending Sellers' result, Das et al. recently proved some congruence results for generalized overcubic partitions using theta functions and posed some related conjectures. In this paper, we provide a combinatorial proof of a result in modulo $4$ of Das et al. and confirm some of their conjectures.

Theorems & Definitions (9)

• Theorem 1.1
• Remark 1.2
• Conjecture 1.3
• Theorem 1.4
• Remark 1.5
• proof : Proof of Theorem \ref{['p3thm1']}
• proof : Proof of Equation \ref{['p3cong1']}
• proof : Proof of Equation \ref{['p3cong2']}
• proof : Proof of Equation \ref{['p3cong3']}