From crank to congruences

Abstract

In this paper, we investigate the arithmetic properties of the difference between the number of partitions of a positive integer $n$ with even crank and those with odd crank, denoted $C(n)=c_e(n)-c_o(n)$. Inspired by Ramanujan's classical congruences for the partition function $p(n)$, we establish a Ramanujan-type congruence for $C(n)$, proving that $C(5n+4)\equiv 0(\mathrm{mod}5)$. Further, we study the generating function $\sum _{n=0}^{\infty }a(n)q^n=\frac{(-q;q{)}_{\infty }^2}{(q;q{)}_{\infty }}$, which arises naturally in this context, and provide multiple combinatorial interpretations for the sequence $a(n)$. We then offer a complete characterization of the values $a(n)\mathrm{mod}{2}^m$ for $m=1,2,3,4$, highlighting their connection to generalized pentagonal numbers. Using computational methods and modular forms, we also derive new identities and congruences, including $a(7n+2)\equiv 0(\mathrm{mod}7)$, expanding the scope of partition congruences in arithmetic progressions. These results build upon classical techniques and recent computational advances, revealing deep combinatorial and modular structure within partition functions.

18 pages