In Stanley's Enumerative Combinatorics, there is a question on Chapter $1$ which goes as follows:
Let $S(n,k)$ denote a Stirling number of the kind (ie, $S(n,k)$ is the number of ways to distribute $n$ elements over $k$ non-empty subsets). For each $n$, define $K_n \in \mathbb N$ and $t_n \in \mathbb R$ by
$$S(n,K_n) \geq S(n,k)$$
for all $k \in \mathbb N$ and
$$\frac{(t_n+2)t_n\log(t_n+2)}{t_n+1} = n$$
Show that for all sufficiently large $n$, either $K_n= \lfloor t_n \rfloor$ or $K_n= \lfloor t_n \rfloor + 1$.
I admit that I've been truly at loss on where to start with. I can't seem to relate $t_n$'s equation to $S(n,k)$'s definition or its properties (recurrence relation, closed form expression, etc).
All help is appreciated!
EDIT: Just clarifying that $\lfloor t_n\rfloor$ is the floor function, which returns the greatest integer that is less than or equal to $t_n$.