Young diagrams of (very close to) maximal dimension     misc

This visualization displays the Young diagrams with the maximum dimension (number of standard Young tableaux) or close to maximum (for large $n$). Here $n$ is the number of boxes in the Young diagram. For large $n$, Young diagrams maximizing $f^\lambda$ are identified via heuristics inspired by this paper by Duzhin and Smirnov-Maltsev (2023). All data on this page was precomputed with various degree of precision (and thus, closeness to the maximal). Up to $n=500$, this should likely be the correct maximal dimension for most $n$ (with a few outliers which are hard to catch), and after that, the answer is approximate, but should be reasonably close.

In our search, we build candidate Young diagrams at level $n+1$ recursively by adding boxes to "best bets" from the previous two levels (few dozen at level $n-1$, and few hundred at level $n$). We also add "shaking": after adding boxes, we allow to move up to $k$ additional boxes in the Young diagram. We maximize the dimension over all candidates, and store the result and the "best bets" for future use.

The algorithm's parameters differ significantly between small and large $n$. For $n\le 500$, we implement larger pools of "best bets", and more extensive shaking. For $500<n \le 5000$, we allow to move only one box. After $n=5000$, we implement an even faster greedy algorithm which just maximizes over all ways to add a box to the previous partition, without shaking.

Conjecture. We believe that even the simplest greedy algorithm hits the actual maximal dimension for infinitely many $n$.

Observation. All partitions up to $n=12000$ included in the current dataset are either symmetric, or have all their asymmetry boxes (the skew diagram $\lambda/(\lambda\Delta\lambda')$, where $\lambda\Delta\lambda'$ is the maximal symmetric subdiagram of $\lambda$, and $\lambda'$ is the conjugate of $\lambda$) exclusively below the diagonal. Moreover, in every row or column there is at most one asymmetry box.