A bijection for partitions with all ranks at least \(t\)
type: Item
instance of: scholarly article
title: A bijection for partitions with all ranks at least \(t\)
zbMATH Open document ID: 0914.05003
author: Sylvie Corteel
author: Carla D. Savage
author: Radhika Venkatraman
published in: Journal of Combinatorial Theory. Series A
publication date: 1999-06-21T00:00:00Z
review text: A nonincreasing sequence \(\lambda= (\lambda_1,\dots, \lambda_t)\) with all its `parts' \(\lambda_i\) positive integers and such that the weight \(|\lambda|= \lambda_1+\cdots+ \lambda_t\) of \(\lambda\) is \(n\), is called a partition of the natural number \(n\) and this fact is denoted by \(\lambda\vdash n\). For a given partition \(\lambda\) let \(\lambda^*= (\lambda^*_1,\dots, \lambda^*_s)\) be its conjugate partition (i.e., \(s= \lambda_1\) and \(\lambda^*_j\) is defined as the number of dots in the \(j\)th column of the Ferrers diagram for \(\lambda\)) and let \(d= d(\lambda)\) denote the length of a side of the Durfee square in the Ferrers diagram for \(\lambda\). The Durfee rectangle of a partition \(\lambda\) is defined to be the largest \(d^*\times (d^*+1)\)-rectangle contained in the Ferrers diagram of \(\lambda\); here \(d^*\) is called the height of the Durfee rectangle and denoted by \(d^*(\lambda)\). According to \textit{H. Gupta} [Fibonacci Q. 16, 548-552 (1978; Zbl 0399.10017)] the rank vector of \(\lambda\) is defined as the vector \(\overline r(\lambda)= (r_1,\dots, r_d)\) with \(r_i= \lambda_i- \lambda^*_i\) \((i= 1,\dots, d)\); the components \(r_i\) are called successive ranks of \(\lambda\). \({\mathfrak P}(n)\) denote the set of all partitions of \(n\), \({\mathfrak P}_b(n)\) the set of partitions with no part equal to \(b\), and \({\mathfrak R}(n)\) the set of those partitions in \({\mathfrak P}(n)\) that have all their successive ranks positive. It was observed by \textit{P. Erdős} and \textit{L. B. Richmond} [Combinatorica 13, 57-63 (1993; Zbl 0790.05008)] that the conjugate to any partition \(\lambda\) in \({\mathfrak R}(n)\) is graphical, i.e., \(\lambda\) is the degree sequence of some simple (undirected) graph. So \(|{\mathfrak R}(n)|\) serves for a lower bound on the number of graphical partitions \(\lambda\), \(\lambda\vdash n\). It has been known earlier that the results of \textit{G. E. Andrews} [Theory arithmetic functions, Lecture…
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 05C99
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 1212978
zbMATH Keywords: Ferrers diagram
zbMATH Keywords: Durfee rectangle
zbMATH Keywords: degree sequence
zbMATH Keywords: graphical partitions
zbMATH Keywords: Rogers-Ramanujan identities
zbMATH Keywords: Durfee square
zbMATH Keywords: generating functions
zbMATH Keywords: set partitions
reviewed by: U. Kaljulaid
MaRDI profile type: Publication
full work available at URL: jcta.1998.2873
OpenAlex ID: W1966143260
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DOI: 10.1006/JCTA.1998.2873
description: scientific article; zbMATH DE number 1212978