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on Tableaux problem 5

The statement

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 $\lambda$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.