Frobenius partitions and the combinatorics of Ramanujan's \(_{1}\psi_{1}\) summation
type: Item
instance of: scholarly article
title: Frobenius partitions and the combinatorics of Ramanujan's \(_{1}\psi_{1}\) summation
zbMATH Open document ID: 0998.11050
author: Sylvie Corteel
author: Jeremy Lovejoy
published in: Journal of Combinatorial Theory. Series A
publication date: 2002-07-04T00:00:00Z
review text: A partition \(\pi\) of \(n\) is a nonincreasing sequence of natural numbers whose sum is \(n\). Let \(p_{_{A,B}}(n)\) denote the number of generalized Frobenius partitions of \(n\), that is, the number of \(2\)-rowed arrays \[ \left(\begin{matrix} a_1 & a_2 & \cdots & a_m \\ b_1 & b_2 & \cdots & b_m \end{matrix}\right), \] where \(\sum a_i\) is a part of type \(A\) and \(\sum b_i\) is a part of type \(B\) and \[ n=\sum (a_i+b_i+1). \] Let \(C\) denote the ``distinct non-negative parts and unrestricted non-negative overlined parts with \(n>\overline{n}\)''. Let \(f_{r,s}(n)\) be the number of generalized Frobenius partitions counted by \(p_{_{C,C}}(n)\) where there are \(r\) overlined parts in the top row and \(s\) overlined parts in the bottom row. Next, let \(g_{r,s}(n)\) be the number of 4-tuples of partitions \((\pi_1,\pi_2,\pi_3,\pi_4)\) where \(\pi_1\) and \(\pi_4\) are ordinary partitions, \(\pi_2\) and \(\pi_3\) are partitions into distinct parts satisfying \[ \sigma(\pi_1)+\sigma(\pi_2)+\sigma(\pi_3)+\sigma(\pi_4)=n, \] \(r=\mu(\pi_2)+\mu(\pi_3)\) and \(s=\mu(\pi_2)+\mu(\pi_4).\) (Here, \(\sigma(\pi)\) denotes the sum of integers in the sequence \(\pi\) and \(\mu(\pi)\) denotes the number of parts in \(\pi\).) In this article, the authors establish a one to one correspondence between the two sets of partitions of \(n\) described above and show that \(f_{r,s}(n)=g_{r,s}(n)\). They show that their result is equivalent to Ramanujan's \(_1\psi_1\) summation formula. The reviewer wishes to point out that the \(\beta\) given in their example (p. 179) should be \[ \beta = \left(\begin{matrix} \overline{5} & 4 & 3 & \overline{3} & \overline{3} & 2 & 1 & \overline{1} & 0 \\ -4 & -2 & -1 & 5 & \overline{5} & \overline{2}& \overline{2}& \overline{2}& \overline{2}\end{matrix}\right). \]
reviewed by: Heng Huat Chan
Mathematics Subject Classification ID: 11P81
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 33D15
zbMATH DE Number: 1763800
zbMATH Keywords: Frobenius partitions
zbMATH Keywords: Ramanujan's \(_1\psi_1\) summation
MaRDI profile type: Publication
full work available at URL: jcta.2001.3205
OpenAlex ID: W2049572756
cites work: Chapter 16 of Ramanujan’s second notebook: theta-functions and 𝑞-series
cites work: Shorter Notes: On Ramanujan's Summation of 1 ψ 1 (a; b; z)
cites work: On a Transformation of Bilateral Series with Applications
cites work: Generalized Frobenius partitions
cites work: A simple proof of Ramanujan's summation of the \(_1\Psi_1\)
cites work: Beiträge zur Theorie der Heineschen Reihen. Die 24 Integrale der hypergeometrischen q‐Differenzengleichung. Das q‐Analogon der Laplace‐Transformation
cites work: Shorter Notes: A Simple Proof of Ramanujan's 1 Ψ 1 Sum
cites work: On Lerch's Transcendant and the Basic Bilateral Hypergeometric Series 2 ψ2
cites work: A Probabilistic Proof of Ramanujan’s ${}_1 \psi _1 $ Sum
DOI: 10.1006/JCTA.2001.3205
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description: scientific article; zbMATH DE number 1763800