Young's lattice and rank functions of differential posets

Abstract

We discuss some conjectures for rank functions of differential posets, and show that these conjectures hold for the Young`s lattice and its Cartesian products. Miller and Stanley conjectured that any differential poset has the nondecreasing property of 1st difference and the nonnegative property of $t$-th difference. Moreover, the inequality ${p_{n + 1}} \le r{p_n} + {p_{n - 1}}$ is the other conjecture raised by Stanley, where $p_n$ is the number of elements in an $r$-differential poset of rank $n$. In this thesis, we show that this inequality establishs for the Young`s lattice and its Cartesian products by constructing injections.