Dyson’s partition ranks and their multiplicative extensions

Hou, Elaine; Jagadeesan, Meena

doi:10.1007/s11139-016-9881-2

Cited by 27 publications

(18 citation statements)

References 9 publications

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“…We note that unlike in [9] our proof of Theorem 1.3 does not give an explicit lower bound on a and b. To yield such a bound one could employ similar techniques to those in [9], relying on the asymptotics found in [2]. However, since [2] gives results only for odd t one could only find such bounds directly for odd t. Further, to find an explicit bound for general t is a difficult problem.…”

mentioning

confidence: 99%

Asymptotic equidistribution and convexity for partition ranks

Males

2020

Ramanujan J

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We study the Dyson rank function N (r, t; n), the number of partitions with rank congruent to r modulo t. We first show that it is monotonic in n, and then show that it equidistributed as n → ∞. Using this result we prove a conjecture of Hou and Jagadeeson on the convexity of N (r, t; n). acknowledgementsThe author would like to thank Kathrin Bringmann for helpful comments on previous versions of the paper, as well as Chris Jennings-Shaffer for useful conversations. The author would also like to thank the referee for many helpful comments.Recently, in [1] Ono and Bessenrodt showed that the partition function satisfies the following convexity result. If a, b ≥ 1 and a + b ≥ 9 then p(a)p(b) > p(a + b).A natural question to ask is then: does N (r, t; n) satisfy a similar property? In [9] Hou and Jagadeesan provide an answer if t = 3. They showed that for 0 ≤ r ≤ 2 we have N (r, 3; a)N (r, 3; b) > N (r, 3; a + b) for all a, b larger than some specific bound. Further, at the end of the same paper, the authors offer the following conjecture on a more general convexity result. Conjecture 1.2. For 0 ≤ r < t and t ≥ 2 then N (r, t; a)N (r, t; b) > N (r, t; a + b) for sufficiently large a and b.As a simple consequence of Theorem 1.1 we prove the following theorem. Theorem 1.3. Conjecture 1.2 is true.Remark. We note that unlike in [9] our proof of Theorem 1.3 does not give an explicit lower bound on a and b. To yield such a bound one could employ similar techniques to those in [9], relying on the asymptotics found in [2]. However, since [2] gives results only for odd t one could only find such bounds directly for odd t. Further, to find an explicit bound for general t is a difficult problem.

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confidence: 99%

Asymptotic equidistribution and convexity for partition ranks

Males

2020

Ramanujan J

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“…A similar phenomenon for partition ranks congruent to a (mod b), denoted by N (a, b; n), was investigated by Hou and Jagadeeson [29], who gave an explicit lower bound on n for convexity of N (a, 2; n). Confirming a conjecture of [29], the third author showed in [36] that for large enough n 1 , n 2 we have…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 64%

Asymptotic equidistribution for partition statistics and topological invariants

Giulia¹,

Craig²,

Males³

2021

Preprint

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We provide a general framework for proving asymptotic equidistribution, convexity, and logconcavity of coefficients of generating functions on arithmetic progressions. Our central tool is a variant of Wright's Circle Method proved by two of the authors with Bringmann and Ono, following work of Ngo and Rhoades. We offer a selection of different examples of such results, proving asymptotic equidistribution results for several partition statistics, modular sums of Betti numbers of two-and three-flag Hilbert schemes, and the number of cells of dimension a (mod b) of a certain scheme central in work of Göttsche.

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“…Bessenrodt-Ono type inequalities appeared also in works by Beckwith and Bessenrodt [2] on k-regular partitions and Hou and Jagadeesan [15] on the numbers of partitions with ranks in a given residue class modulo 3. Males [18] obtained results for general t and Dawsey and Masri [7] obtained new results for the Andrews spt-function.…”

Section: Introduction and Main Resultsmentioning

confidence: 89%

Proof of the Bessenrodt–Ono Inequality by Induction

Heim

Neuhäuser

2021

Res. number theory

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In 2016 Bessenrodt–Ono discovered an inequality addressing additive and multiplicative properties of the partition function. Generalizations by several authors have been given; on partitions with rank in a given residue class by Hou–Jagadeesan and Males, on k-regular partitions by Beckwith–Bessenrodt, on k-colored partitions by Chern–Fu–Tang, and Heim–Neuhauser on their polynomization, and Dawsey–Masri on the Andrews spt-function. The proofs depend on non-trivial asymptotic formulas related to the circle method on one side, or a sophisticated combinatorial proof invented by Alanazi–Gagola–Munagi. We offer in this paper a new proof of the Bessenrodt–Ono inequality, which is built on a well-known recursion formula for partition numbers. We extend the proof to the result by Chern–Fu–Tang and its polynomization. Finally, we also obtain a new result.

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Dyson’s partition ranks and their multiplicative extensions

Cited by 27 publications

References 9 publications

Asymptotic equidistribution and convexity for partition ranks

Asymptotic equidistribution and convexity for partition ranks

Asymptotic equidistribution for partition statistics and topological invariants

Proof of the Bessenrodt–Ono Inequality by Induction

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