Specifically, I am interested only in partitions with at least the amount of $2$ in each bucket, but intuitively it doesn't matter?
I have no idea how to work with those complex asymptotic growth formulas. Looking for a simple approximation such as: looks exponential, or looks logarithmic.
With each part at least 2, is equivalent to "without any part =1", and is given in http://oeis.org/A002865. GF: Product_{m>1} 1/(1-x^m) recursion :a(n)= p(n)-p(n-1) and a(n) ~ Pi * exp(sqrt(2*n/3)*Pi) / (12*sqrt(2)*n^(3/2))