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This is a Quantitative Study study.

September 30, 2025Open Access

From crank to congruences

Key Points

C(n) represents the difference in partitions with even and odd crank values, revealing intriguing modular properties.
Establishing the congruence C(5n+4) ≡ 0 (mod 5) connects to foundational results in partition theory.
Generating functions lead to combinatorial interpretations, illustrating the profound relationship of sequences a(n) with pentagonal numbers.
Computational methods demonstrate new identities and congruences, notably a(7n+2) ≡ 0 (mod 7), enhancing partition congruence literature.

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) =cₑ (n) -cₒ (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) 0 5. Further, we study the generating function ₍=₀^ a (n) \, qⁿ = (-q; q) ² (ₐ; ₐ) _, 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) 2ᵐ 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) 0 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.

From crank to congruences

Key Points

Abstract

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