Let $n,L$ are positive integers. $(m_1,\cdots,m_k)$ is partition of $n$, i.e. $m_1+\cdots+m_k=n$ or $(m_1,\cdots,m_k) \vdash n$.
Note that a partition of n is a representation of n as a sum of positive integers where the order of the summands is considered irrelevant. Thus 2 + 1 + 1, 1 + 2 + 1 and 1 + 1 + 2 are three distinct compositions, but are all considered to be the same partition of 5.
A partition $(m_1,\cdots,m_k) \vdash n$ can be written in multiplicity form $p_1^{a_1} \cdots p_l^{a_l}$. For example, $(2,3,3)\vdash 8:=2+2\times 3$.
My question is, what condition on $\beta(n)$ such that the following quantity decay exponentially with $n$, for example, ~$\exp(-cn)$ where $c$ is constant does not depend on $n$? Note that $\beta(n)$ should be an increasing function of $n$.
\begin{equation} \sum_{\substack{p_1^{a_1} \cdots p_l^{a_l}\vdash n \\ k\leq L}}\binom{L}{k}k!\frac{n!}{p_1^{a_1}\cdot \cdots\cdot p_l^{a_l}\cdot a_1!\cdots a_l!}\cdot\frac{1}{((p_1-1)!)^{a_1}\cdot \cdots \cdot ((p_l-1)!)^{a_l}}\cdot \\ \exp(-\frac{\binom{n}{2}-\big( a_1\binom{p_1}{2}+\cdots+a_l\binom{p_l}{2} \big)}{2\beta})\end{equation} where $k:=a_1+\cdots+a_l$.
Note that the nominator $\binom{n}{2}-\big( a_1\binom{p_1}{2}+\cdots+a_l\binom{p_l}{2} \big)$ inside of $\exp$ can also be written as $\binom{n}{2}-\big( \binom{m_1}{2}+\cdots+\binom{m_k}{2} \big)$.
A backup: if such $\beta(n)$ does not exist then we may consider $L$ as an increasing function of $n$ if this would help.