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A periodic approach to plane partition congruences

Matthew S. Mizuhara, James A. Sellers, Holly Swisher

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.

A periodic approach to plane partition congruences

Abstract

Ramanujan's celebrated congruences of the partition function 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.

Paper Structure

This paper contains 9 sections, 9 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Partitions and Plane Partitions
  3. Plane Partition Congruences
  4. Preliminaries
  5. Proof of Theorem \ref{['congtheorem']}
  6. Known Congruences
  7. Congruences involving prime powers
  8. Plane partition congruences related to multipartition congruences
  9. Conclusion

Key Result

Theorem \oldthetheorem

Fix positive integers $s,t$ and nonnegative integers $a_i,\,b_j$ for each $1\leq i\leq s, 1\leq j \leq t$. For a prime $\ell$, if holds for all $n< \frac{\pi_{\ell}(F_\ell)}{\ell}$, then it holds for all $n\geq 0$.

Theorems & Definitions (16)

  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem: Kwong KWOPart
  • Corollary \oldthetheorem
  • Lemma \oldthetheorem
  • Theorem \oldthetheorem
  • Remark
  • Remark
  • ...and 6 more