3
$\begingroup$

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.

$\endgroup$
2
  • 1
    $\begingroup$ What is the definition of "Hook number"? $\endgroup$ Apr 4 at 22:23
  • $\begingroup$ Added a definition $\endgroup$
    – Mr Lolo
    Apr 4 at 22:33

1 Answer 1

2
$\begingroup$

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:

# 
# 
# # 
# # # 
# # # 
# # # # 
# # # # # # 
# # # # # # # 
# # # # # # # # # 
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.