I'm interested in the limit
$$ L:= \lim_{n \to \infty} \left(\frac{\lceil an \rceil ! \lfloor n-an \rfloor!}{\lceil bn \rceil! \lfloor n-bn\rfloor!} \hspace{.3cm} \right)^{\frac{1}{n}} $$where $0<a,b<1/2$ and $\lceil \cdot \rceil$, $\lfloor \cdot \rfloor$ are the ceiling and floor operators respectively.
Clearly when $a=b$, $L=1$, but in general, my graphing calculator suggests that $L$ is always positive and bounded, and greater than one iff $a>b$.
How should I approach an analytical (dis)proof of this?