complete estimates of the error for a well-known asymptotic expression of partition p(n,m) - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T10:06:16Z http://mathoverflow.net/feeds/question/37360 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37360/complete-estimates-of-the-error-for-a-well-known-asymptotic-expression-of-partiti complete estimates of the error for a well-known asymptotic expression of partition p(n,m) QHLIU 2010-09-01T09:43:21Z 2010-09-02T04:14:36Z <p>Let $p(n,m)$ be the number of partitions of an integer $n$ into integers $\le m$, we have a well-known asymptotic expression: </p> <p>For a fixed $m$ and $n\to\infty$, $$p(n,m)=\frac{n^{m-1}}{m!(m-1)!} (1+O(1/n))$$</p> <p>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. </p> http://mathoverflow.net/questions/37360/complete-estimates-of-the-error-for-a-well-known-asymptotic-expression-of-partiti/37364#37364 Answer by QHLIU for complete estimates of the error for a well-known asymptotic expression of partition p(n,m) QHLIU 2010-09-01T10:38:43Z 2010-09-01T10:49:42Z <p>Thank Robin Chapman very much for editing.</p> <p>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 $n\to\infty$,</p> <p>$$q(n,M)\approx \frac{(n-1)!}{M!(M-1)!(n-M)!}\left( 1+O\left( \frac{M^{3}}{n} \right) \right)$$</p> <p>Isn't there no similar asymptotic expression for partition $p(n,m)$?</p> http://mathoverflow.net/questions/37360/complete-estimates-of-the-error-for-a-well-known-asymptotic-expression-of-partiti/37368#37368 Answer by Gjergji Zaimi for complete estimates of the error for a well-known asymptotic expression of partition p(n,m) Gjergji Zaimi 2010-09-01T11:28:14Z 2010-09-01T11:43:44Z <p>I'm not entirely sure of what you are asking, but note that Erdos and Lehner proved <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.dmj/1077492649" rel="nofollow">here</a> 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 $$p(n,A)=\frac{1}{\prod_{a\in A}a}\frac{n^{m-1}}{(m-1)!}+O(n^{m-2}).$$</p> <p>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.</p> http://mathoverflow.net/questions/37360/complete-estimates-of-the-error-for-a-well-known-asymptotic-expression-of-partiti/37453#37453 Answer by Richard Stanley for complete estimates of the error for a well-known asymptotic expression of partition p(n,m) Richard Stanley 2010-09-02T02:21:50Z 2010-09-02T04:14:36Z <p><a href="http://qjmath.oxfordjournals.org/content/4/1/96.full.pdf+html" rel="nofollow">G. Szekeres, <em>Quart. J. Math. (Oxford)</em> <strong>4</strong>(2) (1953), 96-111</a>, obtains an asymptotic formula for $p(n,m)$ valid uniformly for all $n$ and $m$.</p>