A New Companion to Capparelli's Identities

TL;DR

This paper introduces a new combinatorial interpretation related to Capparelli's identities, extends existing generating functions, and proposes conjectures on inequalities, enriching the understanding of partition identities.

Contribution

It presents a novel partition set equinumerous with Capparelli's partitions, extends generating functions for refined partitions, and formulates new conjectures on combinatorial inequalities.

Findings

New partition set equinumerous with Capparelli's identities.

Extended generating functions for refined Capparelli partitions.

Proposed conjectures on combinatorial inequalities.

Abstract

We discuss a new companion to Capparelli's identities. Capparelli's identities for m=1,2 state that the number of partitions of $n$ into distinct parts not congruent to m, -m modulo $6$ is equal to the number of partitions of n into distinct parts not equal to m, where the difference between parts is greater than or equal to 4, unless consecutive parts are either both consecutive multiples of 3 or add up to to a multiple of 6. In this paper we show that the set of partitions of n into distinct parts where the odd-indexed parts are not congruent to m modulo 3, the even-indexed parts are not congruent to -m modulo 3, and 3l+1 and 3l+2 do not appear together as consecutive parts for any integer l has the same number of elements as the above mentioned Capparelli's partitions of n. In this study we also extend the work of Alladi, Andrews and Gordon by providing a complete set of generating functions for the refined Capparelli partitions, and conjecture some combinatorial inequalities.