Abstract
We study extremal problems concerning the Möbius function μ of certain families of subsets from On, the lattice of faces of the n-dimensional octahedron. For lower order ideals F from On, |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 Bn and On, we demonstrate that the alternating ranks are the only extremal configuration for these two face latties.
Original language | English |
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Pages (from-to) | 361-380 |
Number of pages | 20 |
Journal | Discrete Mathematics |
Volume | 139 |
Issue number | 1-3 |
DOIs | |
State | Published - May 24 1995 |
Keywords
- Convex polytopes
- Cubical lattice
- Extremal configuration
- Lattice of faces
- Lower order ideals
- Maximum
- Möbius function
- Octahedron
- Rank selections
- cd-index
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics