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- W4381846401 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 $$Delta _d^{(3,-)}(n)ge 0$$ for all $$dge 1$$ and $$nge 1$$ and were able to prove this when $$d ge 31$$ is divisible by 3. They were also able to conjecture an analog for higher values of a that the modified difference function $$Delta _{d}^{(a,-,-)}(n) = q_{d}^{(a)}(n) - Q_{d}^{(a,-,-)}(n) ge 0$$ 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 $$Delta _{d}^{(3,-)}(n) ge 0$$ 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 et al. We provide a conditional linear lower bound on d for the generalized conjecture by using a variant of Alder’s conjecture. Additionally, we obtain asymptotic evidence that this modification holds for sufficiently large n." @default.
- W4381846401 created "2023-06-25" @default.
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- W4381846401 date "2023-06-23" @default.
- W4381846401 modified "2023-09-27" @default.
- W4381846401 title "On generalizations of a conjecture of Kang and Park" @default.
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- W4381846401 doi "https://doi.org/10.1007/s40993-023-00459-5" @default.
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