A refined view of a curious identity for partitions into odd parts with designated summands

2025-05-27

AI Summary

This paper investigates the integer partition function PDO(n), which counts partitions of n into odd parts with a specified addend. Addressing a long-standing open problem concerning its modulo-4 congruence properties, the authors derive a novel q-series identity for PDO(n), revealing a decomposable modular-form structure of its generating function. Employing combinatorial bijections, q-series analysis, and symbolic computation, they fully characterize the modulo-4 distribution of PDO(n), thereby proving several previously conjectured congruences. Moreover, they establish the first nontrivial identity for PDO(n) endowed with a combinatorial interpretation. This work advances the understanding of odd-part symmetry in constrained partitions and introduces new methodological tools—integrating modular forms, bijective combinatorics, and computational verification—for analyzing congruence properties of partition functions.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Non-convex Optimization
• Application Domains: Humanities & Computational Social Science

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Federated Web and WoT systems, including distributed, federated and edge-based data processing
• Security and Privacy: Blockchains and distributed ledgers

Abstract

In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called partitions with designated summands. These are constructed by taking unrestricted integer partitions and designating exactly one of each occurrence of a part. In the same work, they also considered the restricted partitions with designated summands wherein all parts must be odd, and they denoted the corresponding function by $mathrm{PDO}(n)$.

Problem

• Study partitions with designated summands for odd parts
• Analyze the function PDO(n) for restricted partitions
• Explore combinatorial properties of designated summand partitions

Innovation

• Designated summands in unrestricted partitions
• Restricted partitions with odd parts
• Combinatorial function PDO(n) analysis