Let $L$ be a constant, $n$ be positive integer, $p_1^{a_1}\cdots p_l^{a_l}$ be a partition of integer $n$.

Define

$$f(L, a_1,\cdots,a_l,p_1,\cdots,p_l,n):=\binom{L}{\sum_{i=1}^l a_i}(\sum_{i=1}^l a_i)!\frac{n!}{(p_1!)^{a_1}\cdots(p_l!)^{a_l}a_1!\cdots a_l!}$$

My question is, how to compute or find the upper bound of

\begin{equation} \sum_{\substack{n^1\neq p_1^{a_1} \cdots p_l^{a_l}\vdash n, \\ \sum_{i=1}^la_i\leq L},\\ \sum_{i=1}^l \binom{p_i}{2}a_i=N(n)}f(L, a_1,\cdots,a_l,p_1,\cdots,p_l,n) \end{equation} where $N(n)$ is a function of $n$.

If for general $n$, it is hard to analyze this formula, it is also nice to consider asymptotic analysis.

I have no experience working with this type of summation, and have zero knowledge in enumerative combinatorics/integer partitions. Thus I am not sure whether this problem is too hard or has some hope.

I would appreciate for any comment!