On the lines-planes inequality for matroids
type: Item
instance of: scholarly article
title: On the lines-planes inequality for matroids
zbMATH Open document ID: 0959.05023
author: Joseph P. S. Kung
published in: Journal of Combinatorial Theory. Series A
publication date: 2001-04-20T00:00:00Z
review text: Let \(G\) be a combinatorial geometry (or simple matroid). Let \(W_i\) be the number of rank-\(i\) flats in \(G\), so \(W_1\) is the number of points, \(W_2\) the number of lines, and so on. Rota's famous unimodality conjecture states that if \(G\) is a rank-\(n\) geometry, then there is a rank \(s\) such that \(W_0\leq W_1 \leq W_2 \leq \cdots \leq W_s\) and \(W_{s+1}\geq W_{s+2}\geq \cdots \geq W_n\). The author proves a partial result concerning the case \(n=5\). Namely, it is shown here that if \(G\) is a geometry of rank at least 5 in which all the lines have the same number of points, then \(W_2 \leq W_3\). The basic idea is to use a special case in a result of \textit{T. A. Dowling} and \textit{R. M. Wilson} [Proc. Am. Math. Soc. 47, 504-512 (1975; Zbl 0297.05010)] together with an explicit ``reconstruction'' algorithm developed by the author in [Math Proc. Camb. Philos. Soc. 101, 221-231 (1987; Zbl 0626.06008)].
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 1525314
zbMATH Keywords: simple matroid
zbMATH Keywords: unimodality conjecture
zbMATH Keywords: Radon transform
reviewed by: Gary L. Ebert
MaRDI profile type: Publication
full work available at URL: jcta.2000.3101
OpenAlex ID: W2004720099
cites work: Q5619114
cites work: On the Foundations of Combinatorial Theory. VII: Symmetric Functions through the Theory of Distribution and Occupancy
cites work: Whitney Number Inequalities for Geometric Lattices
cites work: Q5642570
cites work: Q3726128
cites work: Matchings and Radon transforms in lattices II. Concordant sets
cites work: On the foundations of combinatorial theory I. Theory of M�bius Functions
DOI: 10.1006/JCTA.2000.3101
description: scientific article; zbMATH DE number 1525314