On partitions with initial repetitions
type: Item
instance of: scholarly article
title: On partitions with initial repetitions
zbMATH Open document ID: 1395.05020
author: Augustine Munagi
author: Darlison Nyirenda
published in: The Ramanujan Journal
publication date: 2018-07-20T00:00:00Z
review text: \textit{G. E. Andrews} [Acta Math. Sin., Engl. Ser. 25, No. 9, 1437--1442 (2009; Zbl 1189.05018)] introduced partitions with initial repetitions and established a theorem showing that partitions of \(n\) with initial \(k\)-repetitions are equinumerous to partitions of \(n\) in which no part has multiplicity exceeding \(2k-1\). His proof used generating functions. Later, \textit{W. J. Keith} [Ramanujan J. 27, No. 2, 163--167 (2012; Zbl 1243.05030)] provided a bijective proof by means of \(k\)-modular diagrams. The present paper gives a simplified bijection by showing that a certain decomposition of a partition can be obtained by algebraic operations instead of the \(k\)-modular diagrams. In a second part of the paper, the authors obtain several corollaries and generalizations of Andrews' theorem [loc. cit.] including certain theorems on partitions with ``early conditions'' as defined by \textit{G. E. Andrews} [in: Advances in combinatorics. In part based on the 3rd Waterloo workshop on computer algebra (WWCA, W80) 2011, Waterloo, Canada, May 26--29, 2011. Dedicated to Herbert Saul Wilf on the occasion of his 80th birthday. Berlin: Springer. 57--76 (2013; Zbl 1271.05014)]. Finally, they reinterpret several formulas from Slater's 1952 list by means of certain extensions of partitions with initial repetitions.
reviewed by: Stefano Capparelli
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 11P84
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 6908973
zbMATH Keywords: partition
zbMATH Keywords: partition identity
zbMATH Keywords: bijection
zbMATH Keywords: generating function
MaRDI profile type: Publication
full work available at URL: s11139-017-9896-3
OpenAlex ID: W2611592440
cites work: Rogers-Ramanujan identities for n-color partitions
cites work: A bijection for partitions with initial repetitions
cites work: Partitions with initial repetitions
cites work: Partitions with Early Conditions
cites work: Bijective proofs using two-line matrix representations for partitions
cites work: Further Identities of the Rogers-Ramanujan Type
DOI: 10.1007/S11139-017-9896-3
description: scientific article; zbMATH DE number 6908973