Asymptotic equidistribution and convexity for partition ranks

Abstract

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.

Asymptotic equidistribution for partition statistics and topological invariants

“…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…”

“…Recently, there has been a body of work in analogy with Dirichlet's theorem on the asymptotic equidistribution (or non-equidistribution) on arithmetic progressions of various objects. For example, the third author showed the asymptotic equidistribution of the partition ranks in [36], Ciolan proved asymptotic equidistribution results for the number of partitions of n into k-th powers in [16] and Zhou proved asymptotic equidistribution of a wide class of partition objects in [52].…”

Asymptotic distribution of odd balanced unimodal sequences with rank congruent to a modulo c

“…Results of this type were explored by Ciolan for the overpartition function [8] and by Males for the partition function [14].…”

Proof of the Bessenrodt–Ono Inequality by Induction

“…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. The authors of this paper generalized the Chern-Fu-Tang Theorem to D'Arcais polynomials, also known as Nekrasov-Okounkov polynomials [6,10,11,14,19,20].…”