Structural aspects of differential posets

Abstract

This thesis investigates some structural properties of differential posets and answers some open questions. There are three main results. First, a proof is given that the largest rank size of the nth rank of an r-differential poset is given by the nth rank of the Fibonacci r-differential poset. This solves a question of Stanley. Second, a proof is given that the only 1-differential lattices are Young’s lattice and the Fibonacci 1-differential poset. This also solves a question of Stanley. Third, it is shown that any quantized r-differential poset has the Fibonacci rdifferential poset as its underlying r-differential poset. This negatively answers a question of Lam. Further, a method for computing all partial 1-differential posets up to a given rank is described. The results of this computation up to rank 9 are also included.