Some results concerning partitions with designated summands
type: Item
instance of: scholarly article
title: Some results concerning partitions with designated summands
zbMATH Open document ID: 1468.11215
author: Shane Chern
author: Michael D. Hirschhorn
published in: The Ramanujan Journal
publication date: 2021-05-06T00:00:00Z
review text: The notion of partitions with designated summands was introduced by \textit{G. E.~Andrews} et al. [Acta Arith. 105, No. 1, 51--66 (2002; Zbl 1004.05006)]. Let \(\mathrm{PD}(n)\) and \(\mathrm{PDO}(n)\) denote respectively, the number of partitions of \(n\) with designated summands and the number of partitions of \(n\) with designated summands where all parts are odd. They derived some Ramanujan-type identities and congruences for \(\mathrm{PD}(n)\) and \(\mathrm{PDO}(n)\). For example, \begin{align*} \sum_{n=0}^\infty\mathrm{PD}(2n+1)q^n &=\frac{(q^2;q^2)_\infty^5(q^{12};q^{12})_\infty^2} {(q;q)_\infty^5(q^3;q^3)_\infty(q^4;q^4)_\infty^2(q^6;q^6)_\infty},\tag{1}\\ \sum_{n=0}^\infty\mathrm{PDO}(3n+2)q^n &=\frac{2(q^2;q^2)_\infty^3(q^6;q^6)_\infty (q^{12};q^{12})_\infty}{(q;q)_\infty^4(q^4;q^4)_\infty}.\tag{2} \end{align*} In 2013, \textit{W.Y.C.~Chen} et al. [J. Number Theory 133, No. 9, 2929--2938 (2013; Zbl 1295.05044)] further studied combinatorial and congruence properties for \(\mathrm{PD}(n)\). Later, \textit{E.X.W.~Xia} [J. Number Theory 159, 160--175 (2016; Zbl 1330.11066)] proved some infinite families of congruences modulo small powers of \(3\) for \(\mathrm{PD}(n)\). Recently, \textit{B.L.S.~Lin} [J. Number Theory 184, 216--234 (2018; Zbl 1375.05021)] introduced two new functions \(\mathrm{PD}_\mathrm{t}(n)\) and \(\mathrm{PDO}_\mathrm{t}(n)\), which denote the number of tags (designated summands) in the partitions enumerated, respectively, by \(\mathrm{PD}(n)\) and \(\mathrm{PDO}(n)\). He derived that \begin{align*} \sum_{n=0}^\infty\mathrm{PD}_\mathrm{t}(n)q^n &=\frac{1}{2}{\left(\frac{(q^3;q^3)_\infty^5} {(q;q)_\infty^3(q^6;q^6)_\infty^2}-\frac{(q^6;q^6)_\infty} {(q;q)_\infty(q^2;q^2)_\infty(q^3;q^3)_\infty}\right)} \end{align*} and \begin{align*} \sum_{n=0}^\infty\mathrm{PDO}_\mathrm{t}(n)q^n &=\frac{q(q^2;q^2)_\infty(q^3;q^3)_\infty^2 (q^{12};q^{12})_\infty^2}{(q;q)_\infty^2(q^6;q^6)_\infty}. \end{align*} Lin also obtained many congruences for \(\math…
reviewed by: Dazhao Tang
Mathematics Subject Classification ID: 11P83
Mathematics Subject Classification ID: 05A17
zbMATH DE Number: 7344961
zbMATH Keywords: partitions with designated summands
zbMATH Keywords: tagged parts
zbMATH Keywords: congruences
MaRDI profile type: Publication
full work available at URL: s11139-019-00198-1
OpenAlex ID: W3003745727
cites work: New infinite families of congruences for the number of tagged parts over partitions with designated summands
cites work: Partitions with designated summands
cites work: On the number of partitions with designated summands
cites work: The power of \(q\). A personal journey
cites work: The number of tagged parts over the partitions with designated summands
cites work: Arithmetic properties of partitions with designated summands
DOI: 10.1007/S11139-019-00198-1
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description: scientific article; zbMATH DE number 7344961