On the lattice of order ideals of an up-down poset

Abstract

The combinatorial structure of the distributive lattice of order ideals of an up-down poset is studied. Two recursions are given for the Whitney numbers, and generating functions for the Whitney numbers are derived. In addition, an explicit nested chain decomposition is given for the lattice, the existence of which implies that the lattice satisfies the Sperner property and its generalizations, and has unimodal Whitney numbers.