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Let's say we have some shape $\lambda$ and we want to fill this shape with numbers $\{1, .., m\}$ in non-decreasing order in rows and columns. How many such numberings do we have? I can not find anything about this kind of numbering of Young tableaux.

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What is the exact requirement to the order of numbering and what requirements does $\lambda$ meet? – AlexR Jul 23 '13 at 14:29
I suggest you to google "Robinson-Schensted-Knuth correspondence". There is also a very good book by Fulton on Young tableaux which may be of some help. – Start wearing purple Jul 23 '13 at 14:33

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