Tableaux problem 5

5.[Wor] Consider the process of randomly choosing a Young diagram of size $n$. If we let $n\rightarrow\infty$ and scale the diagram down by $\sqrt{n}$ in both dimensions, the average of the diagrams converges to a limit shape; points within the shape have probability 1 of being in the random diagram and points outside the shape have probability 0. Surprisingly, the same limit shape results under three distributions on the set of diagrams: uniform (weight $1$), uniform on Young tableaux (weight $f_{\lambda}$, the number of tableaux of shape $\lambda$), and the Plancherel measure (weight $f_{\lambda}^{2}$).

Perform the parallel analysis on shifted Young diagrams, the set of partitions into distinct parts. Since the RSK algorithm for shifted diagrams has a weighting ($c_{\lambda}=2^{|\lambda|-\ell(\lambda)}$), there are now six plausible distributions: $1$, $g_{\lambda}$, $g_{\lambda}^{2}$, $c_{\lambda}$, $g_{\lambda}c_{\lambda}$, and $g_{\lambda}^{2}c_{\lambda}$, where $g_{\lambda}$ is the number of shifted Young tableaux (without coloring) of shape $\lambda$.