Extremal problems for the Möbius function in the face lattice of the n-octahedron

Abstract

We study extremal problems concerning the Möbius function μ of certain families of subsets from O n , the lattice of faces of the n-dimensional octahedron. For lower order ideals F from O n , |μ( F)| attains a unique maximum by taking F to be the lower two-thirds of the ranks of the poset. Stanley showed that the coefficients of the cd-index for face lattices of convex polytopes are non-negative. We verify an observation that this result implies that the Möbius function is maximized over arbitrary rank-selections from these lattices by taking their odd or even ranks. Using recurrences by Purtill for the cd-index of B n and O n , we demonstrate that the alternating ranks are the only extremal configuration for these two face latties.