On ranks and cranks of partitions modulo $4$ and $8$

TL;DR

This paper provides new proofs and insights into the properties of ranks and cranks of partitions modulo 4, 5, 7, and 8, connecting them with mock theta functions and establishing new identities and inequalities.

Contribution

It introduces novel proofs of existing results and new identities involving ranks and cranks of partitions, especially for moduli 4, 5, 7, and 8, using deviation functions.

Findings

New proofs of recent results on mock theta functions and partition ranks.

New identities and inequalities for rank-crank differences, especially for modulus 8.

Results on cranks for moduli 5 and 7, with some appearing to be new.

Abstract

Denote by $p(n)$ the number of partitions of $n$ and by $N(a,M;n)$ the number of partitions of $n$ with rank congruent to $a$ modulo $M$. By considering the deviation \begin{equation*} D(a,M) := \sum_{n= 0}^{\infty}\left(N(a,M;n) - \frac{p(n)}{M}\right) q^n, \end{equation*} we give new proofs of recent results of Andrews, Berndt, Chan, Kim and Malik on mock theta functions and ranks of partitions. By considering deviations of cranks, we give new proofs of Lewis and Santa-Gadea's rank-crank identities. We revisit ranks and cranks modulus $M=5$ and $7$, with our results on cranks appearing to be new. We also demonstrate how considering deviations of ranks and cranks gives first proofs of Lewis's conjectured identities and inequalities for rank-crank differences of modulus $M=8$.