Given $L$ as an absolute constant (does not depend on $n$), and $\beta$ is a function of $n$. Consider
$$\sum_{m_1+\cdots+m_k=n\\ 1\leq k\leq L\\ \text{all }m_i \text{ are positive integers}}\binom{L}{k}k!\binom{n}{m_1,\cdots,m_k}\exp(-\frac{\binom{n}{2}-(\binom{m_1}{2}+\cdots+\binom{m_k}{2})}{2\beta})$$
where $\binom{1}{2}$ is set as $0$.
Is it possible to find the condition of $\beta(n)$ (it has to be positive and increasing with $n$), such that the above summation over all partitions (the order of $(m_1,\cdots,m_k)$ does not matter) of $n$, is exponentially decaying when $n$ grows, for example, ~$\exp(-cn)$ where $c$ is certain absolute constant.