A partition of $n$ is a weakly decreasing tuple of numbers $(\lambda\_1,\lambda\_2,\lambda\_3,....\lambda\_k)$ whose sum is $n.$ A natural problem studied is counting partitions whose summands $\lambda\_i$ all like in a set $S\subset \mathbb{N}$ and we denote this number $p\_S(n)$. 


If $S$ is ``dense" enough, one can usually calculate an asymptotic formula that looks like $$p\_S(n) = Kn^\beta\exp\left(c\sqrt{n}\right)\left(1+ \frac{c\_1}{\sqrt{n}}+ \frac{c\_2}{n}+\frac{c\_3}{n^\frac{3}{2}} \dots \right).$$


My question is when $S=\mathbb{N}\setminus [0,1,2,\dots c]$ does such a formula already exist in the literature and where can I find it?



Does the asymptotic formula for Partitions into parts <c exist?
Parts greater than $c$ is complementary to parts smaller than $c.$ (as pointed out in the comments, parts greater than $c$ is the same as "at least $c$ parts") for the latter, there is a theorem of Szekeres from 1951, see the [math review.](https://dl.dropboxusercontent.com/u/5188175/szekeres.pdf) (of. Course this doesn't answer the question, but will leave answer up for the seekers reference) 2 3rd 


