1
$\begingroup$

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?

$\endgroup$
2
  • $\begingroup$ 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. $\endgroup$ Jul 15, 2013 at 3:29
  • $\begingroup$ I forgot the decreasing part of the definition and its parts greater than c rather than less than c. My fault and I apologize. $\endgroup$ Jul 15, 2013 at 3:36

1 Answer 1

1
$\begingroup$

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

$\endgroup$
1

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.