Maximum area of rectangle in a young tableau. Given a positive integer $n$, is there a formula or a bound on the maximum area of rectangles contained in the Young diagrams of the partitions of $n$.
For example when $n=5$, the Young diagrams for the partitions of $5$ are
$\Box\Box\Box\Box\Box$
$\Box\Box\Box\Box$$\Box$
$\Box\Box\Box$$\Box\Box$
$\Box\Box\Box$$\Box$$\Box$
$\Box\Box$$\Box\Box$$\Box$
$\Box\Box$$\Box$$\Box$$\Box$
$\Box$$\Box$$\Box$$\Box$$\Box$
which contain a rectangle of maximum area $5,4,4,3,4,4,5$ respectively. So the quantity I want for when $n=5$ is $3$.

Consider the rectangular hyperbola $xy=A$ in the quadrant $x, y \gt 0$.
Any rectangle below the hyperbola has area $\le A$, and clearly the tightest hyperbola to any configuration passes through at least one corner, so that for the limiting case there is a rectangle of area $A$.
The question then is how to find the minimum value of $A$ for a given value of $n$. The number of squares which fit under a such hyperbola is given by
$$n=\sum_{i=1}^{\lfloor A \rfloor}\lfloor \frac Ai\rfloor\sim A(\ln A +\gamma)$$
Others will do the estimates better than me to get a bound, but it looks as though $A\sim \cfrac n {\ln n}$.
