I am trying to find the $k$ that maximizes of $f:\mathbb{Z}_{+}\to\mathbb{R}$ given by $f(k)= k^{-n} \frac{k!}{(k-m)!} \mathbf{1}_{k\geq m}$ as a function of $n$ and $m$, both positive integers with $m<n$. I tried replacing the factorial with the stadard asymptotic approximation and maximizing the resulting function. However, finding the zero of the derivative amounts to solving an equation involving $k^2, k$ and $\log(k)$, which is not possible to do exactly. I am wondering if there might be some easier trick to get the maximum that I might be overlooking ?
As the function has only one local/global maximum, the argmax is also given by the first $k$ for which the ratio $f(k)\big/f(k+1)= \big(1+\frac{1}{k}\big)^n \big(1- \frac{m}{k+1} \big)$ is larger than $1$, meaning that right after that point the function starts to decrease.