Congruence properties modulo powers of 2 for two partition functions
type: Item
instance of: scholarly article
title: Congruence properties modulo powers of 2 for two partition functions
author: Dazhao Tang
published in: Journal of the Ramanujan Mathematical Society
publication date: 2024-10-07T00:00:00Z
review text: 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.
reviewed by: Donna Q. J. Dou
Mathematics Subject Classification ID: 11P83
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 05A15
zbMATH DE Number: 7922355
zbMATH Keywords: partitions into distinct parts
zbMATH Keywords: 1-shell totally symmetric plane partitions
MaRDI profile type: Publication
cites work: Plane partitions. VI: Stembridge's TSPP theorem
cites work: Proof of a conjecture of Ramanujan
cites work: Exact generating functions for the number of partitions into distinct parts
cites work: Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary
cites work: Q4898146
cites work: Congruences for 1-shell totally symmetric plane partitions
cites work: 1-shell totally symmetric plane partitions (TSPPs) modulo powers of 5
cites work: Partitions into distinct parts modulo powers of 5
cites work: THE ROGERS–RAMANUJAN CONTINUED FRACTION AND RELATED ETA-QUOTIENT REPRESENTATIONS
cites work: Q3929810
cites work: Divisibility of certain partition functions by powers of primes
cites work: The power of \(q\). A personal journey
cites work: ARITHMETIC PROPERTIES OF 1-SHELL TOTALLY SYMMETRIC PLANE PARTITIONS
cites work: Overpartition pairs modulo powers of 2
cites work: THE NUMBER OF PARTITIONS INTO DISTINCT PARTS MODULO POWERS OF 5
cites work: The overpartition function modulo small powers of 2
cites work: The 2-adic behavior of the number of partitions into distinct parts
cites work: Q5654122
cites work: On the computation of identities relating partition numbers in arithmetic progressions with eta quotients: an implementation of Radu's algorithm
cites work: The enumeration of totally symmetric plane partitions
cites work: Congruence properties modulo powers of 2 for overpartitions and overpartition pairs
cites work: Ramanujans Vermutung über Zerfällungsanzahlen.
cites work: A NEW CONGRUENCE MODULO 25 FOR 1-SHELL TOTALLY SYMMETRIC PLANE PARTITIONS
cites work: NEW INFINITE FAMILIES OF CONGRUENCES MODULO 4 AND 8 FOR 1-SHELL TOTALLY SYMMETRIC PLANE PARTITIONS
description: scientific article; zbMATH DE number 7922355