The ranks and cranks of partitions moduli 2, 3, and 4
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.
scientific article; zbMATH DE number 1547374