# Given n, do we have a formula for the greatest Hook number of an n-box Young diagram?

Obviously the least Hook number is 1, by considering boxes stacked vertically or horizontally. Is there a formula for the greatest possible Hook number ?

EDIT: The Hook number of a Young diagram is defined to be the number of standard Young tableaux whose shape is that of the Young diagram.

• What is the definition of "Hook number"? Apr 4 at 22:23
• Added a definition Apr 4 at 22:33

Too long for a comment, but I wrote some Python code to compute the partition(s) of $$n$$ which attain the maximal hook number. It is brute force, so it is only feasible for sufficiently small $$n$$, say $$n\le 50$$. Link to code.

Here are the optimal partitions for $$n$$ between $$28$$ and $$36$$. There seems to always be exactly one or two partitions attaining the maximum. The optimal partitions are also roughly triangular, even though the triangular partition $$(m,m-1,\dots,2,1)$$ is itself not optimal except for small enough $$m$$.

n = 28:

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n = 29:

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n = 30:

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n = 31:

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n = 32:

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n = 33:

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n = 34:

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n = 35:

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n = 36:

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