Let $C_n$ denote subset of integer compositions of $n$ and $c=(c_1,c_2,\dots c_n)$

In a divergent sum, the sequence $$ a_n=\sum_{c\in C_n} \prod_{c_i\in c} c_i! $$ frequently shows up and one expects it to be asymptotic to the contributions containing its maximal part. I just want to know if this kind of integer composition has been studied and if I can find some explicit estimates/bounds for it that already exist in the literature.