complete estimates of the error for a well-known asymptotic expression of partition p(n,m) - MathOverflow

complete estimates of the error for a well-known asymptotic expression of partition p(n,m)

2010-09-01T09:43:21Z

Let $p(n,m)$ be the number of partitions of an integer $n$ into integers $\le m$, we have a well-known asymptotic expression:

For a fixed $m$ and $n\to\infty$, $$p(n,m)=\frac{n^{m-1}}{m!(m-1)!} (1+O(1/n)) $$

My question is: why the error $O(1/n)$ is independent of $m$? Or how can it be extended for $m$ growing slowly with $n$? Please help me to find the answer or the references. Thanks.

Answer by QHLIU for complete estimates of the error for a well-known asymptotic expression of partition p(n,m)

2010-09-01T10:38:43Z

Thank Robin Chapman very much for editing.

There is a nice asymptotic expression for partition $q(n,M)$ that denotes the number of partitions of $n$ with $M$ parts all distinct: As $n\to\infty$,

$$ q(n,M)\approx \frac{(n-1)!}{M!(M-1)!(n-M)!}\left( 1+O\left( \frac{M^{3}}{n} \right) \right)$$

Isn't there no similar asymptotic expression for partition $p(n,m)$?

Answer by Gjergji Zaimi for complete estimates of the error for a well-known asymptotic expression of partition p(n,m)

2010-09-01T11:28:14Z

I'm not entirely sure of what you are asking, but note that Erdos and Lehner proved here that $$p(n,m)\sim \frac{n^{m-1}}{m!(m-1)!}$$ holds for $m=o(n^{1/3})$. In generality for any finite set $A$, with $|A|=m$ and $p(n,A)$ denoting the number of partitions of $n$ with parts from $A$, one has $$p(n,A)=\frac{1}{\prod_{a\in A}a}\frac{n^{m-1}}{(m-1)!}+O(n^{m-2}).$$

Such estimations can be deduced from the generating function of $p$ by using methods that are described in many books, for example "Analytic Combinatorics" by Flajolet and Sedgewick.

Answer by Richard Stanley for complete estimates of the error for a well-known asymptotic expression of partition p(n,m)

2010-09-02T02:21:50Z

G. Szekeres, Quart. J. Math. (Oxford) 4(2) (1953), 96-111, obtains an asymptotic formula for $p(n,m)$ valid uniformly for all $n$ and $m$.