The ranks and cranks of partitions moduli 2, 3, and 4
type: Item
instance of: scholarly article
title: The ranks and cranks of partitions moduli 2, 3, and 4
zbMATH Open document ID: 0967.11043
author: George E. Andrews
author: Richard P. Lewis
published in: Journal of Number Theory
publication date: 2001-01-01T00:00:00Z
review text: Let \(N(r,m,n)\) be the number of partitions of \(n\) with rank congruent to \(r\) modulo \(m\) and let \(M(r,m,n)\) be the number of partitions of \(n\) with crank congruent to \(r\) modulo \(m\). For fixed \(n\) and \(m = 2, 3,\) or \(4\), the authors give theorems and conjectures for inequalities between the values of the rank at different values of \(r\) and the values of the crank at different values of \(r\). For example, a conjecture that appears to be related to the Borwein conjecture [\textit{G. E. Andrews}, J. Symb. Comput. 20, 487--501 (1995; Zbl 0849.68062)] is that for all \(n\), \[ \begin{aligned} M(0,3,3n) & > M(1,3,3n), \\ M(0,3,3n+1) & < M(1,3,3n+1), \\ M(0,3,3n+2) & \leq M(1,3,3n+2), \quad n \neq 1. \end{aligned} \] {}. Remark: The first author's name is misspelled in the original as George A. Andrews.
Mathematics Subject Classification ID: 11P83
zbMATH DE Number: 1547374
zbMATH Keywords: partitions
zbMATH Keywords: rank
zbMATH Keywords: crank
zbMATH Keywords: Borwein conjecture
reviewed by: David M. Bressoud
MaRDI profile type: Publication
OpenAlex ID: W2040154089
cites work: Q4146776
cites work: On a conjecture of Peter Borwein
cites work: Ramanujan's ``Lost'' Notebook. VII: The sixth order mock theta functions
cites work: Some Properties of Partitions
cites work: Inequalities for the crank
cites work: Q3797261
cites work: New Combinatorial Interpretations of Ramanujan's Partition Congruences Mod 5,7 and 11
cites work: The ranks of partitions modulo 2
cites work: The Final Problem : An Account of the Mock Theta Functions
DOI: 10.1006/JNTH.2000.2537
description: scientific article; zbMATH DE number 1547374