A congruence family for 2-elongated plane partitions: an application of the localization method
type: Item
instance of: scholarly article
title: A congruence family for 2-elongated plane partitions: an application of the localization method
author: Nicolas Allen Smoot
published in: Journal of Number Theory
publication date: 2022-09-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/2111.07131
review text: For a nonnegative integer \(k\), \textit{G. E. Andrews} and \textit{P. Paule} [J. Number Theory 234, 95--119 (2022; Zbl 1484.11201)] considered the \(k\)-elongated plane partition function \(d_k(n)\) where \[ \sum_{n=0}^{\infty}d_k(n)q^n=\prod_{n=1}^{\infty} \frac {(1-q^{2n})^k}{(1-q^n)^{3k+1}}, \quad |q|<1. \] \(d_k(n)\) generalizes the unrestricted partition function since \(d_0(n)=p(n)\). Moreover, \(k\)-elongated plane partitions are also generalizations of partition diamonds developed by \textit{G. E. Andrews} et al. [Adv. Appl. Math. 27, 231--242 (2001; Zbl 0992.05017)]. Andrews and Paule [loc. cit.] conjectured that, for positive integers \(n\) and \(\alpha\), \(8n\equiv 1 \bmod 3^{\alpha}\) implies that \(d_2(n)\equiv 0 \bmod 3^{\alpha}\). In the paper under review the author proves the refined form \(d_2(n)\equiv 0 \bmod 3^{2\lfloor \alpha/2\rfloor+1}\) of the conjecture by expressing the associated generating functions as elements of a ring of modular functions isomorphic to a localization of \(\mathbb{Z}[X]\).
Mathematics Subject Classification ID: 11P83
Mathematics Subject Classification ID: 30F35
zbMATH DE Number: 7595055
zbMATH Keywords: partition congruences
zbMATH Keywords: modular functions
zbMATH Keywords: plane partitions
zbMATH Keywords: partition analysis
zbMATH Keywords: localization method
zbMATH Keywords: modular curve
zbMATH Keywords: Riemann surface
reviewed by: Mihály Szalay
Wikidata QID: Q114156407
MaRDI profile type: Publication
OpenAlex ID: W3212464593
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DOI: 10.1016/J.JNT.2022.07.014
description: scientific article; zbMATH DE number 7595055