✍ Scribed by Margaret A. Readdy
No coin nor oath required. For personal study only.
We study extremal problems concerning the M6bius function kt of certain families of subsets from 0,, the lattice of faces of the n-dimensional octahedron. For lower order ideals ff from O,, tP(~}] attains a unique maximum by taking ff 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 M6bius 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, and O,, we demonstrate that the alternating ranks are the only extremal configuration for these two face latties.
📜 SIMILAR VOLUMES
Erdős has conjectured that every subgraph of the n-cube Q n having more than (1/2+o(1))e(Q n ) edges will contain a 4-cycle. In this note we consider 'layer' graphs, namely, subgraphs of the cube spanned by the subsets of sizes k -1, k and k + 1, where we are thinking of the vertices of Q n as being
## Abstract The main theme of this paper is the discussion of a parametrized family of solutions of a finite moment problem for rational matrix‐valued functions in the nondegenerate case. We will show that each member of this family is extremal in several directions concerning some point of the ope