An inequality between finite analogues of rank and crank moments

The inequality between rank and crank moments was conjectured and later proved by Garvan himself in 2011. Recently, Dixit and the authors introduced finite analogues of rank and crank moments for vector partitions while deriving a finite analogue of Andrews’ famous identity for smallest parts function. In the same paper, they also conjectured an inequality between finite analogues of rank and crank moments, analogous to Garvan’s conjecture. In this paper, we give a proof of this conjecture.

Newman's identities, lucas sequences and congruences for certain partition functions

Let r be an integer with 2 ≤ r ≤ 24 and let pr(n) be defined by $\sum _{n=0}^\infty p_r(n) q^n = \prod _{k=1}^\infty (1-q^k)^r$. In this paper, we provide uniform methods for discovering infinite families of congruences and strange congruences for pr(n) by using some identities on pr(n) due to Newman. As applications, we establish many infinite families of congruences and strange congruences for certain partition functions, such as Andrews's smallest parts function, the coefficients of Ramanujan's ϕ function and p-regular partition functions. For example, we prove that for n ≥ 0, \[ \textrm{spt}\bigg( \frac{1991n(3n+1) }{2} +83\bigg) \equiv \textrm{spt}\bigg(\frac{1991n(3n+5)}{2} +2074\bigg) \equiv 0\ (\textrm{mod} \ 11), \] and for k ≥ 0, \[ \textrm{spt}\bigg( \frac{143\times 5^{6k} +1 }{24}\bigg)\equiv 2^{k+2} \ (\textrm{mod}\ 11), \] where spt(n) denotes Andrews's smallest parts function.

Effective bounds for the Andrews spt-function

In this paper, we establish an asymptotic formula with an effective bound on the error term for the Andrews smallest parts function {{\mathrm{spt}}(n)} . We use this formula to prove recent conjectures of Chen concerning inequalities which involve the partition function {p(n)} and {{\mathrm{spt}}(n)} . Further, we strengthen one of the conjectures, and prove that for every {\epsilon>0} there is an effectively computable constant {N(\epsilon)>0} such that for all {n\geq N(\epsilon)} , we have \frac{\sqrt{6}}{\pi}\sqrt{n}\,p(n)<{\mathrm{spt}}(n)<\bigg{(}\frac{\sqrt{6}}{% \pi}+\epsilon\bigg{)}\sqrt{n}\,p(n). Due to the conditional convergence of the Rademacher-type formula for {{\mathrm{spt}}(n)} , we must employ methods which are completely different from those used by Lehmer to give effective error bounds for {p(n)} . Instead, our approach relies on the fact that {p(n)} and {{\mathrm{spt}}(n)} can be expressed as traces of singular moduli.

Congruences for some partitions related to mock theta functions

Partitions related to mock theta functions were widely studied in the literature. Recently, Andrews et al. introduced two new kinds of partitions counted by [Formula: see text] and [Formula: see text], whose generating functions are [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are two third mock theta functions. Meanwhile, they obtained some congruences for [Formula: see text], [Formula: see text], and the associated smallest parts function [Formula: see text]. Furthermore, Andrews et al. discussed the overpartition analogues of [Formula: see text] and [Formula: see text] which are denoted by [Formula: see text] and [Formula: see text]. In this paper, we derive more congruences for [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. Moreover, we establish some congruences for [Formula: see text] and its associated smallest parts function [Formula: see text], where [Formula: see text] denotes the number of overpartitions of [Formula: see text] such that all even parts are at most twice the smallest part, and in which the smallest part is always overlined.

A strange partition theorem related to the second Atkin–Garvan moment

This paper contains results on a strange smallest parts function related to the second Atkin–Garvan moment. Some new identities are discovered in relation to Andrews spt function as well as one of Borweins' two-dimensional theta functions.

HECKE-TYPE CONGRUENCES FOR TWO SMALLEST PARTS FUNCTIONS

We prove infinitely many congruences modulo 3, 5, and powers of 2 for the overpartition function [Formula: see text] and two smallest parts functions: [Formula: see text] for overpartitions and M2spt(n) for partitions without repeated odd parts. These resemble the Hecke-type congruences found by Atkin for the partition function p(n) in 1966 and Garvan for the smallest parts function spt(n) in 2010. The proofs depend on congruences between the generating functions for [Formula: see text], [Formula: see text], and M2spt(n) and eigenforms for the half-integral weight Hecke operator T(ℓ2).