Let $p(n)$ denote the partitions of $n$. It's easy to prove that $p(n) < n!$ (for n>2). I want to prove that if $n \ge 6$ then $(n-2)! > p(n)$. Or more generally let $k_0\in \mathbb{N}$ be a fixed positive integer and suppose that for some $n_0 \in \mathbb{N}$ $(n_0-k_0)! > p(n_0)$. Then $(n-k_0)! > p(n)$ for each $n \ge n_0$.
I don't know if this it's possible by simple methods because the recursive formulas for $p(n)$ are very complicated.