Is anything known about the following question? Given a positive integer $p$ and a real number $0<\alpha<1$, what partition $\lambda$ whose parts sum to $\alpha p^2$ (asymptotically) and whose diagram fits in a $p\times p$ square maximizes the number of semistandard Young tableaux of shape $\lambda$ and largest part at most $p$? In particular, is there a limiting curve like there is for the partition $\lambda$ of $n$ maximizing the number of standard Young tableaux of shape $\lambda$? By the FrameRobinsonThrall hook length formula, this latter question is related to the variational problem $$ \min\int_A \log(f(x)+f^{1}(y)xy)dx\,dy, $$ over all nonincreasing nonnegative $f$ on $(0,\infty)$ satisfying $\int_0^\infty f(x)dx =1$, and where $A$ is the region between the $x$axis and the graph of $f(x)$. See for instance http://stat.wharton.upenn.edu/~shepp/publications/55.pdf. For the first question, I believe that the corresponding variational problem is by the hookcontent formula $$ \max\int_A (\log(1+xy)\log(f(x)+f^{1}(y)xy))dx\,dy, $$ where $f\colon (0,1)\to(0,1)$, $f$ is nonincreasing, and $\int_0^1 f(x)dx = \alpha$. A number of similar questions can be formulated.
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$\begingroup$ Interesting, this would also, in the limit, maximize the volume of the corresponding GelfandTsetlin polytope, with boundary values given by the restriction on the partitions... $\endgroup$ – Per Alexandersson Mar 3 '15 at 21:37

$\begingroup$ I am almost sure that the limit curve exists, may be calculated less or more explicitly, and the maximal diagram is also near this curve. $\endgroup$ – Fedor Petrov Mar 3 '15 at 23:33