Extremal problems for the Möbius function in the face lattice of the \(n\)-octahedron
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 \(O_n\).
scientific article; zbMATH DE number 773918
scientific article