Thursday, December 6, 2018
All day 

Before 1am
1am
2am
3am
4am
5am
6am
7am
8am
9am
10am
11am
12pm
1pm
2pm
3pm
4pm
5pm
6pm
7pm
8pm
9pm
10pm
11pm

12/06/2018  4:00pm We find an asymptotic formula for the number of partitions of n whose Young diagrams fit inside an m by l rectangle, equivalently, the coefficients of the qbinomial coefficient ${m + l \choose m}$. Our formula is valid throughout the regime where $l, m$ and $\sqrt{n}$ are all comparable. Previously, sharp asymptotics were derived by Takacs (1986), only in the Central Limit regime, where n is within order $m$ of $l m / 2$. Our approach is to solve a large deviation problem. We describe the tilted measure that produces configurations whose bounding rectangle has the given aspect ratio and is filled to the given proportion. Our results are sufficiently sharp also to yield the first asymptotic estimates on the consecutive differences of these numbers when n is increased by one and m and l remain the same, refining Sylvester’s unimodality theorem. Location:
DL 431
