Extremal problems for the Möbius function in the face lattice of the \(n\)-octahedron
type: Item
instance of: scholarly article
title: Extremal problems for the Möbius function in the face lattice of the \(n\)-octahedron
zbMATH Open document ID: 0827.52006
DOI: 10.1016/0012-365X(94)00141-5
published in: Discrete Mathematics
publication date: 1995-07-13T00:00:00Z
review text: Let \(O_n\) denote the face lattice of the \(n\)-dimensional cross- polytope. In this paper, extremal problems with respect to the Möbius function \(\mu\) of certain families of subsets from the poset \(O_n\) are studied. Recently, \textit{B. E. Sagan}, \textit{Y.-N. Yeh} and \textit{G. M. Ziegler} [Discrete Math. 126, No. 1-3, 293-311 (1994; Zbl 0809.05004)] approached extremal problems involving the Möbius function \(\mu\). As poset under consideration they fixed the Boolean algebra \(B_n\) and investigated the maximum value attained by \(\mu\) over certain subsets of \(B_n\). In the present paper, the techniques of Sagan et al. are extended to \(O_n\), and the extremal configuration for lower order ideals is found to coincide with the lower two-thirds of the poset \(O_n\). It was shown by P. R. Stanley that the coefficients of the \(cd\)-index for face lattices of convex polytopes are nonnegative. Here an observation is verified, namely: this result implies that \(\mu\) is maximized over arbitrary rank-selections from these lattices by taking their odd or even ranks. It is demonstrated that the alternating ranks are the only extremal configuration for the face lattices \(B_n\) and \(O_n\).
reviewed by: Horst Martini
Mathematics Subject Classification ID: 52B05
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 06A07
zbMATH DE Number: 773918
zbMATH Keywords: \(cd\)-index
zbMATH Keywords: face lattice
zbMATH Keywords: cross-polytope
zbMATH Keywords: Möbius function
zbMATH Keywords: poset
author: Margaret A. Readdy
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 773918