Samuel F. Hopkins
Howard University
Title
Combinatorics Seminar: Upho posets
Abstract
A partially ordered set is called upper homogeneous, or “upho,” if every principal order filter is isomorphic to the whole poset. This class of fractal-like posets was recently introduced by Stanley. Our first observation is that the rank generating function of a (finite type N-graded) upho poset is the reciprocal of its characteristic generating function. This means that each upho lattice has associated to it a finite graded lattice, called its core, that determines its rank generating function. With an eye towards classifying upho lattices, we investigate which finite graded lattices arise as cores, providing both positive and negative results. Our overall goal for this talk is to advertise upho posets, and especially upho lattices, which we believe are a natural and rich class of posets deserving of further attention. Essentially no background knowledge will be assumed, and we also hope to highlight several open problems.
