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
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.