Congruence properties modulo powers of 2 for two partition functions
For any nonnegative integer \(n\), let \(Q(n)\) denote the number of partitions of \(n\) into distinct parts, and \(f(n)\) denote the number of 1-shell totally symmetric plane partitions of \(n\). Congruence properties modulo powers of 2 and 5 for \(Q(n)\) and \(f(n)\) have been extensively studied by several scholars. In the paper under review, by utilizing some \(q\)-series identities and iterative computations, the author establishes a number of internal congruences and congruences modulo powers of 2 satisfied by \(Q(n)\) and \(f(n)\). We state the two main results as follows. If \(n\geq 0\) and \(1\leq\alpha \leq 7\), then \[Q(5^{2^{\alpha+1}}n+\frac{5^{2^{\alpha+1}}-1}{24})\equiv (2^{\alpha+1}+1)Q(n)\ (\text{mod}\,2^{\alpha+2}),\] \[f(6\times 5^{2^{\alpha+1}}n+5^{2^{\alpha+1}})\equiv (2^{\alpha+1}+1)f(6n+1)\ (\text{mod}\,2^{\alpha+2}).\] Moreover, for \(1\leq i\leq 4\), \[Q(5^{2^{\alpha+1}}n+\frac{(24i+5)\times 5^{2^{\alpha+1}-1}-1}{24})\equiv 0\ (\text{mod}\,2^{\alpha+2}),\] \[f(6\times 5^{2^{\alpha+1}}n+(6i+5)\times 5^{2^{\alpha+1}-1})\equiv 0\ (\text{mod}\,2^{\alpha+2}).\] The author also conjectures that the above results hold for each positive integer \(\alpha\). The paper closes with some remarks.
1-shell totally symmetric plane partitions
scientific article; zbMATH DE number 7922355
scientific article