# Does the asymptotic formula for Partitions into parts <c exist?

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?

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The tuple notation generally implies an ordering, whereas in partition counting one generally considers $1+2$ and $2+1$ to be the same partition of 3. Partitions into parts not exceeding $c$ is the same as partitions into at most $c$ parts, and I think asymptotically that's $Cn^{c-1}$, which is negligible compared to the full number of partitions. –  Gerry Myerson Jul 15 '13 at 3:29
I forgot the decreasing part of the definition and its parts greater than c rather than less than c. My fault and I apologize. –  Daniel Parry Jul 15 '13 at 3:36

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. (of. Course this doesn't answer the question, but will leave answer up for the seekers reference) 2 3rd