Arithmetic properties of partitions with designated summands
type: Item
instance of: scholarly article
title: Arithmetic properties of partitions with designated summands
zbMATH Open document ID: 1330.11066
author: Ernest X. W. Xia
published in: Journal of Number Theory
publication date: 2015-10-21T00:00:00Z
review text: Let \(PD(n)\) denote the number of partitions of \(n\) with designated summands. \textit{W. Y. C. Chen} et al. [J. Number Theory 133, No. 9, 2929--2938 (2013; Zbl 1295.05044)] established the generating functions \[ \sum_{n \geq 0} PD(3n+2)q^n = 3\frac{f_3^3f_6^3}{f_1^5f_2^3} \] and \[ \sum_{n \geq 0} PD(3n)q^n = \frac{f_3^3}{f_1^5} \left(\frac{f_4^6f_6}{f_2^5f_{12}^2} + 3q^2\frac{f_{12}^6}{f_2f_4^2f_{6}^3}\right), \] where \[ f_k := \prod_{n \geq 1} (1-q^{kn}). \] With these identities as a starting point, the author proves a number of congruences for \(PD(n)\) modulo \(9\) or \(27\). Among them are an exact formula for \(PD(36n+9)\) modulo \(9\) which depends on the factorization of \(8n+2\), families of Ramanujan-type congruences such as \[ PD(2^{6k-1}(12n+1)) \equiv 0 \pmod{9}, \] and a congruence involving primes \(p \equiv 1 \pmod{12}\), \[ PD(13^{9k}(75p+2)) \equiv 0 \pmod{27}. \]
reviewed by: Jeremy Lovejoy
Mathematics Subject Classification ID: 11P83
Mathematics Subject Classification ID: 05A17
zbMATH DE Number: 6497372
zbMATH Keywords: partitions with designated summands
zbMATH Keywords: congruences
zbMATH Keywords: theta function identities
MaRDI profile type: Publication
full work available at URL: j.jnt.2015.06.016
OpenAlex ID: W1438330104
cites work: Partitions with designated summands
cites work: MacMahon's sum-of-divisors functions, Chebyshev polynomials, and quasi-modular forms
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cites work: On the number of partitions with designated summands
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DOI: 10.1016/J.JNT.2015.06.016
description: scientific article; zbMATH DE number 6497372