The 2-adic behavior of the number of partitions into distinct parts
type: Item
instance of: scholarly article
title: The 2-adic behavior of the number of partitions into distinct parts
zbMATH Open document ID: 0972.05003
author: Ken Ono
author: David Penniston
published in: Journal of Combinatorial Theory. Series A
publication date: 2001-11-13T00:00:00Z
review text: Let \(Q(n)\) be the number of partitions of a nonnegative integer into \(n\) distinct parts. In a series of papers in 1996-97 K. Alladi examined the 2-adic behavior of \(Q(n)\) and in 1997 \textit{B. Gordon} and \textit{K. Ono} [Ramanujan J. 1, No. 1, 25-34 (1997; Zbl 0907.11036)] showed that if \(j\) is a positive integer then \(Q(n)\equiv 0\pmod{2^j}\) for a subset of the nonnegative integers \(n\) with arithmetic density one. Here the authors show, using modular forms, that if \(i\not\equiv 0\pmod{2^j}\), then \(\#\{0\leq n\leq X\mid Q(n)\equiv i\pmod{2^j}\}\gg_j\sqrt X/\log X\). A strong form of this can be proved for even \(i\). A rather more complicated statement is proved for \(Q(n)\pmod 8\), using the arithmetic of the ring \(\mathbb{Z}[\sqrt{-6}]\).
Mathematics Subject Classification ID: 11P83
Mathematics Subject Classification ID: 11F33
zbMATH DE Number: 1569470
zbMATH Keywords: partitions
zbMATH Keywords: modular forms
reviewed by: Gerald L. Alexanderson
MaRDI profile type: Publication
OpenAlex ID: W1989096280
cites work: A combinatorial correspondence related to Göllnitz’ (big) partition theorem and applications
cites work: Partition identities involving gaps and weights
cites work: Q4885296
cites work: A trinomial analogue of Bailey's lemma and \(N=2\) superconformal invariance
cites work: Q4374999
cites work: The construction of modular forms as products of transforms of the Dedekind eta function
cites work: Q3996190
cites work: Q5607047
cites work: Formes modulaires de poids $1$
cites work: Divisibility of certain partition functions by powers of primes
cites work: Q4023805
cites work: Q3346401
cites work: Q4083553
cites work: Q4693061
cites work: Q4213374
cites work: Divisibilite de certaines fonctions arithmétiques
cites work: Q5625265
DOI: 10.1006/JCTA.2000.3057
description: scientific article; zbMATH DE number 1569470