From crank to congruences
Tewodros Amdeberhan, Mircea Merca
TL;DR
The paper studies the arithmetic of the crank parity difference $C(n)$ and a companion generating function $a(n)$ arising from $\frac{(-q;q)_\infty^2}{(q;q)_\infty}$, uncovering Ramanujan-type congruences such as $C(5n+4)\equiv0\pmod{5}$ and $a(7n+2)\equiv0\pmod{7}$. The authors develop a blend of $q$-series techniques, quintisection expansions, and modular form methods to derive eta-quotient identities, complete $2^m$-modulo characterizations for $a(n)$ in terms of generalized pentagonal numbers, and modular-function proofs (via RaduRK) of higher-level congruences. They provide modular-analytic proofs (including eta-quotient identities in $M_0(\Gamma_0(10))$ and $M_0(\Gamma_0(14))$) and offer multiple combinatorial interpretations of $a(n)$, connecting distinct partition coloring schemes to the arithmetic of these sequences. Overall, the work extends classical Ramanujan-type congruences into the crank parity framework, illustrates deep modular structure behind partition statistics, and demonstrates the power of computational tools in proving modular identities with arithmetic implications.
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 $C(5n+4) \equiv 0 \pmod{5}$. Further, we study the generating function $\sum\limits_{n=0}^\infty a(n)\, q^n = \frac{(-q; q)^2_\infty}{(q; q)_\infty}$, 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 $a(n) \mod 2^m$ 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 $a(7n+2) \equiv 0 \pmod{7}$, 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.