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I would appreciate if somebody could help me with the following problem

Q: Arrangements of $1,2,..,81$ in an $9 \times 9$ matrix such that each row and each column is increasing. How many possible $X$ ?

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Young
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  • HINT: Try filling it from 1 to 81 and you will see that "almost" every number has exactly two options. – Alex Fish Nov 07 '15 at 11:26
  • http://puzzling.stackexchange.com/questions/23024/increasing-rows-and-columns may be helpful. – Gerry Myerson Nov 07 '15 at 11:49
  • For a $2\times N$ rectangle, I get $1,2,5,14,...$, which might be Catalan numbers $\frac1{n+1}{2n\choose n}$ or the cumulative sum of powers of three. I think it will be Catalan numbers because it is the number of ways of going from $(0,0)$ to $(n,n)$ staying on or above the diagonal. – Empy2 Nov 07 '15 at 12:31
  • This equals the number of paths from $(0,0,0,0,0,0,0,0,0)$ to $(9,9,9,9,9,9,9,9,9)$ while ensuring the coordinates are always a (non-strictly) decreasing sequence. – Empy2 Nov 07 '15 at 12:40

2 Answers2

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Two obvious constraints are that all the other numbers in the rectangle from top left to $x$ are less than $x$ and all the numbers other than $x$ in the bottom right rectangle from $x$ to the corner are greater than $x$.

That gives a maximum and minimum value. Choose a value $x$ in this constrained range - can you fill in the rest of the numbers?

Mark Bennet
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You are counting standard Young tableaux (in the special case of square tableaux). You want the hook-length formula; see also this question.

Community
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Tad
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