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.

We introduce and investigate an infinite family of functions which are shown to have generalised quantum modular properties. We realise their "companions" in the lower half plane both as double Eichler integrals and as non-holomorphic theta functions with coefficients given by double error functions. Further, we view these Eichler integrals in a modular setting as parts of certain weight two indefinite theta series.

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In this paper we investigate a certain eta-theta quotient which appears in the partition function of entanglement entropy. Employing Wright's circle method, we give its bivariate asymptotic profile.Mathematics Subject Classification 2010: 11F50

Recent works at the interface of algebraic combinatorics, algebraic geometry, number theory, and topology have provided new integer-valued invariants on integer partitions. It is natural to consider the distribution of partitions when sorted by these invariants in congruence classes. We consider the prominent situations which arise from extensions of the Nekrasov-Okounkov hook product formula, and from Betti numbers of various Hilbert schemes of n points on C 2 . For the Hilbert schemes, we prove that homology is equidistributed as n → ∞. For t-hooks, we prove distributions which are often not equidistributed. The cases where t ∈ {2, 3} stand out, as there are congruence classes where such counts are zero. To obtain these distributions, we obtain analytic results which are of independent interest. We determine the asymptotics, near roots of unity, of the ubiquitous infinite products(1 − (ξq) n ) and F3(ξ; q) := ∞ n=1 1 − ξ −1 (ξq) n .