On the Combinatorics of Placing Balls into Ordered Bins

In this paper, we use techniques of enumerative combinatorics to study the following problem: we count the number of ways to split $n$ balls into nonempty, ordered bins so that the most crowded bin has exactly $k$ balls. We find closed forms for three of the different cases that can arise: $k > \frac{n}{2}$, $k = \frac{n}{2}$, and when there exists $j < k$ such that $n = 2k + j$. As an immediate result of our proofs, we find a closed form for the number of positive integer solutions to $x_1 + x_2 + \dots + x_{\ell} = n$ with the attained maximum of $\{x_1, x_2, \dots, x_{\ell}\}$ being equal to $k$, when $n$ and $k$ have one of the aforementioned algebraic relationships to each other. The problem is generalized to find a formula that enumerates the total number of ways without specific conditions on $n, \ell, k$. Subsequently, various additional identities and estimates related to this enumeration are proven and interpreted.