Suppose we have $m$ balls and distribute them into $n$ bins. For $i \in \{0,1,...,m\}$, We let $x_i$ be the number of bins that have $i$ balls in them.
We let $x = (x_i)_{i=0}^m \in \mathbb{N}_{+}^{m+1}$ be the histogram of $x_i$. What is the caridinality of the feasible histogram $x$ (i.e., $x \in \mathbb{N}_{+}^{m+1}$ such that $\sum_{i=0}^m x_i = n$ and $\sum_{i=0}^m i \cdot x_i = m$)?
Any classical results there? Specifically, I'm curious if this number grows polynomially or exponentially fast in $m$ and/or $n$. Thanks so much!