Partitions with rounded occurrences and attached parts
type: Item
instance of: scholarly article
title: Partitions with rounded occurrences and attached parts
zbMATH Open document ID: 1226.05041
author: Jeremy Lovejoy
published in: The Ramanujan Journal
publication date: 2011-01-17T00:00:00Z
review text: Define the series \(S_{k,i}(a,x,q)\) as \[ \begin{multlined} S_{k,i}(a,x,q) = \frac{1}{(xq)_\infty}\sum_{n\geq 0}\frac{a^n x^{(k-1)n}q^{(k-1)n^2+(k-i+1)n} (x^2q^2, 1/a; q^2)_n}{(q^2,ax^2q^2; q^2)_n}\\ \left(1+\frac{ax^iq^{(2n+1)i-2n}(1-q^{2n}/a)}{1-ax^2q^{2n+2}}\right),\end{multlined} \] and write \(S_{k,i}(a,x,q)=\sum_{t,m,n} s_{k,i}(t,m,n) a^t x^m q^n\). The author gives a combinatorial interpretation of the coefficients \(s_{k,i}\) as the number of pairs of partitions \((\lambda, \mu)\) of \(n\) satisfying certain conditions too complicated to be stated here. Although the definition of these restrictions is highly involved, the result is surprisingly useful, as the author demonstrates by giving a new proof of a theorem by \textit{D. M. Bressoud} [``A generalization of the Rogers-Ramanujan identities for all moduli,'' J. Comb. Theory, Ser. A 27, 64--68 (1979; Zbl 0416.10009)] and proving a new identity for the number of certain pairs of overpartitions. The proof uses standard computations involving hypergeometric series.
reviewed by: Jan-Christoph Schlage-Puchta
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 11P81
Mathematics Subject Classification ID: 11P84
zbMATH DE Number: 5837834
zbMATH Keywords: partitions
zbMATH Keywords: hypergeometric series
zbMATH Keywords: Rogers-Ramanujan-identities
zbMATH Keywords: overpartition
MaRDI profile type: Publication
cites work: An Analytic Proof of the Rogers-Ramanujan-Gordon Identities
cites work: Some new partition theorems
cites work: Partition theorems related to the Rogers-Ramanujan identities
cites work: A Generalization of the Gollnitz-Gordon Partition Theorems
cites work: ON q-DIFFERENCE EQUATIONS FOR CERTAIN WELL-POISED BASIC HYPERGEOMETRIC SERIES
cites work: A Generalization of the Classical Partition Theorems
cites work: Applications of Basic Hypergeometric Functions
cites work: On the general Rogers-Ramanujan theorem
cites work: On the Alder Polynomials and a New Generalization of the Rogers-Ramanujan Identities
cites work: The hard-hexagon model and Rogers—Ramanujan type identities
cites work: Multiple series Rogers-Ramanujan type identities
cites work: Parity in partition identities
cites work: Roger-Ramanujan type identities for partitions with attached odd parts
cites work: A generalization of the Rogers-Ramanujan identities for all moduli
cites work: Analytic and combinatorial generalizations of the Rogers-Ramanujan identities
cites work: Partition Theorems Related to Some Identities of Rogers and Watson
cites work: An extension to overpartitions of the Rogers-Ramanujan identities for even moduli
cites work: Overpartitions, lattice paths, and Rogers-Ramanujan identities
cites work: Q4040884
cites work: Gordon's theorem for overpartitions.
cites work: Overpartition Theorems of the Rogers–Ramanujan Type
cites work: Overpartition pairs
cites work: Partitions and overpartitions with attached parts
cites work: Overpartition pairs and two classes of basic hypergeometric series
DOI: 10.1007/S11139-009-9189-6
description: scientific article; zbMATH DE number 5837834