Whitney numbers of some geometric lattices
type: Item
instance of: scholarly article
title: Whitney numbers of some geometric lattices
zbMATH Open document ID: 0793.05037
DOI: 10.1016/0097-3165(94)90034-5
author: E. Damiani
author: Francesco Regonati
author: Ottavio M. D'Antona
published in: Journal of Combinatorial Theory. Series A
publication date: 1994-03-24T00:00:00Z
review text: The authors study sequences of supersolvable geometric lattices \(L_ 1,L_ 2,L_ 3,\dots\) which are uniform in the sense that every upper interval of \(\text{rank }k\) in \(L_ n\) is isomorphic to \(L_ k\). They derive simple recursions for the Whitney numbers of the lattices \(L_ n\). Writing these in terms of symmetric functions lead to a log-concavity proof in a case with sufficiently many modular coatoms. This yields a common lattice theoretic framework for the known results in the special case where all \(L_ n\) are modularly complemented; here a complete classification of the uniform sequences follows from \textit{J. Kahn} and \textit{J. P. S. Kung} [Eur. J. Comb. 7, 243-248 (1986; Zbl 0614.05018)]. It is an open problem whether other examples exist.
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 05E05
zbMATH DE Number: 529904
zbMATH Keywords: supersolvable geometric lattices
zbMATH Keywords: Whitney numbers
zbMATH Keywords: symmetric functions
zbMATH Keywords: log-concavity
zbMATH Keywords: modularly complemented
reviewed by: Günter M. Ziegler
MaRDI profile type: Publication
full work available at URL: 0097-3165(94)90034-5
OpenAlex ID: W2093711842
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description: scientific article; zbMATH DE number 529904