Timeline for complete estimates of the error for a well-known asymptotic expression of partition p(n,m)
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 7, 2010 at 15:19 | comment | added | Richard Stanley | I am not an expert in this area, but see combinatorics.org/Volume_4/Abstracts/v4i2r06ab.html. Here Szekeres' asymptotic formula is given that is valid in the range $k\geq n^{1/6}$, and is valid uniformly in the entire range $k\geq 1$ by adding $1/k$ to the big-oh term. See also the article by Romik in Europ. J. Combinatorics 26 (2005), 1-17. As for whether the formula is useful, Szekeres uses it to prove that for $n$ sufficiently large, the sequence $p(n,1), p(n,2),\dots, p(n,n)$ is unimodal, the only known proof of this result. | |
| Sep 4, 2010 at 8:20 | comment | added | QHLIU | Please be noted that in G. Szekeres' two papers in 1951 and 1953, the asymptotic formulae for $p(n,m)$ valid only under strong condition that $m$ is related to $n^2$. In physics, no asymptotic formula not useful unless $m \alpha n$ ($\alpha >0$ and as $m$ is large). | |
| Sep 2, 2010 at 4:23 | comment | added | Richard Borcherds | His formula only seems to be valid for n at most a constant times m<sup>2</sup>. It also involves the solution of a quite nasty looking implicit equation, so I'm not sure how useful it is. | |
| Sep 2, 2010 at 4:14 | history | edited | Richard Borcherds | CC BY-SA 2.5 |
link to paper
|
| Sep 2, 2010 at 2:35 | history | edited | Richard Stanley | CC BY-SA 2.5 |
edited body
|
| Sep 2, 2010 at 2:21 | history | answered | Richard Stanley | CC BY-SA 2.5 |