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

Question

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.

Answer 1

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:

# 
# 
# # 
# # # 
# # # 
# # # # 
# # # # # # 
# # # # # # # # 

n = 29:

# 
# # 
# # # 
# # # # 
# # # # # 
# # # # # # 
# # # # # # # # 

n = 30:

# 
# 
# # 
# # # 
# # # # 
# # # # # 
# # # # # # 
# # # # # # # # 

n = 31:

# 
# 
# # 
# # # 
# # # # 
# # # # # 
# # # # # # 
# # # # # # # # # 

n = 32:

# 
# 
# # 
# # # 
# # # # 
# # # # # 
# # # # # # # 
# # # # # # # # # 

n = 33:

# 
# 
# 
# # 
# # # 
# # # # 
# # # # # 
# # # # # # # 
# # # # # # # # # 

n = 34:

# 
# 
# # 
# # 
# # # 
# # # # 
# # # # # 
# # # # # # # 
# # # # # # # # # 

n = 35:

# 
# 
# # 
# # 
# # # 
# # # # 
# # # # # # 
# # # # # # # 
# # # # # # # # # 

n = 36:

# 
# 
# # 
# # # 
# # # 
# # # # 
# # # # # # 
# # # # # # # 
# # # # # # # # # 

Answer 2

I don't believe there is a formula for this, but you might look through the references in the links below. There are a few related sequences from OEIS which use the equivalent language "highest degree of an irreducible representation of symmetric group $S_n$."

The sequence of counts is given in entry A003040. Entry A117500 gives a choice of partition for each $n$ with the maximal number of Young tableaux; John McKay's 1976 work linked there matches what Mike Earnest found in his answer (or the conjugate partition, which clearly has the same number of tableaux).