0
$\begingroup$

I'm learning about Young Tableaux.The number of standard Young tableaux of size n can can be generated by the recurrence relation:
$a(n)=a(n-1)+(n-1)a(n-2)$
By definition, A standard Young tableau (SYT) is a filling of a Young diagram with the numbers 1, 2, . . . , n so that entries are increasing along rows and columns.
Now I want to fill with n non-distinct numbers, how can I calculate the number of Young Tableaux again?I'm thinking about using Hook-Length formula, but It does not seem work out.

$\endgroup$
3
  • 1
    $\begingroup$ What rule are you using in place of insisting rows/columns are increasing? Also please see math.meta.stackexchange.com/questions/5020 $\endgroup$ Aug 5, 2018 at 14:27
  • $\begingroup$ I think this is called Counting Inverted semistandard Young Tableaux $\endgroup$
    – VN Pikachu
    Aug 5, 2018 at 14:45
  • $\begingroup$ It might help to give a reference to where you are learning about Young tableaux as the description of "entries... increasing along rows and columns" suggests a matrix rather than what is usually considered (an arrangement of contiguous "boxes" or dots, esp. when the alternative name Ferrers diagram is used). $\endgroup$
    – hardmath
    Aug 5, 2018 at 14:47

1 Answer 1

1
$\begingroup$

The count of all Semi-Standard Young Tableaux (SSYT) of size n and maximal element at most n is given in http://oeis.org/A209673. If you find a closed form expression for this, please let us know. The count of SSYT of shape $\lambda\vdash n$ is given by Stanley's Hook Content Formula. The pairs formed by all SSYT and all SYT of the same shape $\lambda\vdash n$ generate all $n^n$ different n-letter words of at most n different letters by the extended RSK-correspondance.

$\endgroup$

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.