A congruence family for 2-elongated plane partitions: an application of the localization method
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 \(\mathbb{Z}[X]\).
scientific article; zbMATH DE number 7595055