A New Companion to Capparelli's Identities
Alexander Berkovich, Ali Kemal Uncu

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 into distinct parts not congruent to m, -m modulo 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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
