Structural Aspects of Differential Posets.

摘要

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 r-differential 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.