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

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 $M=8$.

26 pages. Section 11 now contains proofs conjectures of R. Lewis

Cited by in corpus (3)

• Deviation of the rank and crank modulo 11
• Arithmetic Properties of Odd Ranks and $k$-Marked Odd Durfee Symbols
• Four identities related to third order mock theta functions