On generalizations of a conjecture of Kang and Park

DOI10.1007/S40993-023-00459-5arXiv2206.04842OpenAlexW4381846401WikidataQ122942657 ScholiaQ122942657MaRDI QIDQ6175150

Ryota Inagaki, Ryan Tamura

Publication date: 21 July 2023

Published in: Research in Number Theory (Search for Journal in Brave)

Abstract: Let $Delta_d^{(a,-)}(n) = q_d^{(a)}(n) - Q_d^{(a,-)}(n)$ where $q_d^{(a)}(n)$ counts the number of partitions of $n$ into parts with difference at least $d$ and size at least $a$, and $Q_d^{(a,-)}(n)$ counts the number of partitions into parts $equiv pm a pmod{d + 3}$ excluding the $d+3-a$ part. Motivated by generalizing a conjecture of Kang and Park, Duncan, Khunger, Swisher, and the second author conjectured that $Delta_d^{(3,-)}(n)geq 0$ for all $dgeq 1$ and $ngeq 1$ and were able to prove this when $d geq 31$ is divisible by $3$. They were also able to conjecture an analog for higher values of $a$ that the modified difference function $Delta_{d}^{(a,-,-)}(n) = q_{d}^{(a)}(n) - Q_{d}^{(a,-,-)}(n) geq 0$ where $Q_{d}^{(a,-,-)}(n)$ counts the number of partitions into parts $equiv pm a pmod{d + 3}$ excluding the $a$ and $d+3-a$ parts and proved it for infinitely many classes of $n$ and $d$. We prove that $Delta_{d}^{(3,-)}(n) geq 0$ for all but finitely many $d$. We also provide a proof of the generalized conjecture for all but finitely many $d$ for fixed $a$ and strengthen the results of Duncan, Khunger, Swisher, and the second author. We provide a conditional proof of a linear lower bound on $d$ for the generalized conjecture, which improves our unconditional result based on a conjectural modification of a recently proven conjecture of Alder. Using this modification, we obtain a strengthening of this generalization of Kang and Park's conjecture which remarkably allows $a$ as a part. Additionally, we provide asymptotic evidence that this strengthened conjecture holds.

Full work available at URL: https://arxiv.org/abs/2206.04842

zbMATH Keywords

partitionsRogers-Ramanujan identitiesAlder's conjecture

Mathematics Subject Classification ID

Combinatorial aspects of partitions of integers (05A17) Forms of half-integer weight; nonholomorphic modular forms (11F37) Analytic theory of partitions (11P82) Partition identities; identities of Rogers-Ramanujan type (11P84)

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Related Items (2)

Alder-type partition inequality at the general level ⋮ Polygonal numbers and Rogers-Ramanujan-Gordon theorem

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