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.
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1$\begingroup$ What rule are you using in place of insisting rows/columns are increasing? Also please see math.meta.stackexchange.com/questions/5020 $\endgroup$– Angina SengAug 5, 2018 at 14:27
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$\begingroup$ I think this is called Counting Inverted semistandard Young Tableaux $\endgroup$– VN PikachuAug 5, 2018 at 14:45
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$\begingroup$ 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). $\endgroup$– hardmathAug 5, 2018 at 14:47
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1 Answer
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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.
