The lattice of non-crossing partitions and the Birkhoff-Lewis equations
type: Item
instance of: scholarly article
title: The lattice of non-crossing partitions and the Birkhoff-Lewis equations
zbMATH Open document ID: 0799.05021
published in: European Journal of Combinatorics
publication date: 1994-06-29T00:00:00Z
review text: A matrix associated with the chromatic join of non-crossing partitions, introduced by Tutte, can be used to generalize the Birkhoff-Lewis equations. Studying the determinant of this matrix gives an interesting conjecture (still open) in which the determinant is expressed in terms of some specific polynomials. The matrix can be described in terms of two lattices. The paper restates some results obtained already earlier by Baeza-Yates, Dahab and Younger. These results are concerned with the determinants of four closely related matrices (however, these matrices can be described in terms of only one lattice), and are proved directly in the incidence algebra of the related lattices. The results can be restated in terms of the combinatorial invariants for the lattices. The main result of the paper is the reformulation of the conjecture in terms of the determination of another matrix of which the description can be given in terms of only one lattice (the lattice of non-crossing partitions).
Mathematics Subject Classification ID: 05C15
Mathematics Subject Classification ID: 05A18
zbMATH DE Number: 597749
zbMATH Keywords: Whitney numbers
zbMATH Keywords: non-crossing partitions
zbMATH Keywords: Birkhoff-Lewis equations
zbMATH Keywords: determinant
zbMATH Keywords: matrix
zbMATH Keywords: lattices
author: David M. Jackson
reviewed by: Herman J. Tiersma
MaRDI profile type: Publication
full work available at URL: eujc.1994.1026
OpenAlex ID: W2095446494
DOI: 10.1006/EUJC.1994.1026
description: scientific article; zbMATH DE number 597749