[Paper Review] A Periodic Approach to Plane Partition Congruences

Matthew S. Mizuhara, James A. Sellers+1 more|arXiv (Cornell University)|Jul 8, 2015

Advanced Mathematical Identities12 references3 citations

TL;DR

This paper introduces a periodicity-based method to prove congruences for k-component plane partition functions using Kwong's theorem on rational function periodicity. It establishes two new congruences modulo prime ℓ for plₖ(n) and proves additional results via connections to multipartition functions, offering a novel, self-contained approach to plane partition congruences without relying on modular forms.

ABSTRACT

Ramanujan's celebrated congruences of the partition function $p(n)$ have inspired a vast amount of results on various partition functions. Kwong's work on periodicity of rational polynomial functions yields a general theorem used to establish congruences for restricted plane partitions. This theorem provides a novel proof of several classical congruences and establishes two new congruences. We additionally prove several new congruences which do not fit the scope of the theorem, using only elementary techniques, or a relationship to existing multipartition congruences.

Motivation & Objective

• To develop a new method for proving plane partition congruences using periodicity of rational generating functions.
• To establish new congruences for k-component plane partitions modulo prime ℓ without relying on modular forms or prior multipartition results.
• To extend known congruences by relating plane partition generating functions to multipartition functions via algebraic manipulation.
• To provide a computational framework where verifying congruences reduces to checking a finite number of initial terms.
• To explore the existence and structure of further congruences of the form plₖ(ℓn + a) ≡ 0 mod ℓ.

Proposed method

• Leverages Kwong’s theorem on periodicity of rational functions to reduce the verification of infinite congruences to a finite number of initial cases.
• Uses the generating function PLₖ(q) = ∏_{n=1}^∞ 1/(1−qⁿ)^{min(k,n)} to express k-component plane partitions.
• Applies Lemma 2.5 to reduce generating functions modulo prime ℓ by factoring out high-order terms and analyzing truncated series.
• Employs algebraic manipulation of generating functions modulo 2, particularly factoring (1−qⁿ)^ℓ ≡ (1−q^{nℓ}) mod 2, to simplify coefficients.
• Uses explicit coefficient computation of low-degree terms in (1−q)^a(1−q²)^b⋯ to verify vanishing modulo ℓ for specific exponents.
• Relies on known multipartition congruences (e.g., Kiming and Olsson’s result) to derive new plane partition congruences via algebraic relations between pₖ(n) and plₖ(n).

Experimental results

Research questions

• RQ1 Can periodicity of rational functions be used to prove new plane partition congruences without modular forms?
• RQ2 What is the minimal number of initial terms needed to verify a plane partition congruence modulo prime ℓ?
• RQ3 Are there additional congruences of the form plₖ(ℓn + a) ≡ 0 mod ℓ beyond those already known?
• RQ4 How can the generating function Fₖ(q) be simplified to make coefficient analysis more efficient?
• RQ5 Can the method be generalized to prime powers or other moduli?

Key findings

• Theorem 1.2 establishes that verifying a congruence modulo ℓ for plₖ(ℓn + aᵢ) reduces to checking only the first πₗ(Fₗ)/ℓ terms, enabling finite verification.
• Two new congruences are proven: pl₄(4n+3) ≡ 0 mod 2 and pl₈(8n+5) ≡ 0 mod 2, confirmed via coefficient analysis of truncated generating functions.
• For ℓ=5, the congruences pl₂(5n+3) ≡ 0 mod 5 and pl₂(5n+4) ≡ 0 mod 5 are established using the relation pl₂(n) = p₂(n) − p₂(n−1) and Kiming-Olsson’s multipartition results.
• The method avoids reliance on modular forms and instead uses elementary algebra and periodicity of rational functions to derive congruences.
• Numerical evidence suggests no further congruences of the form plₗ(nℓ + α) ≡ 0 mod ℓ exist for primes ℓ ≤ 113 and all α, indicating the current results may be exhaustive.
• The approach is computationally bounded, and future work may benefit from more efficient forms of Fₖ(q) or generalizations to prime powers.