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A060854
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Array T(m,n) read by antidiagonals: T(m,n) (m >= 1, n >= 1) = number of ways to arrange the numbers 1,2,..,m*n in an m X n matrix so that each row and each column is increasing.
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16
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1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 14, 42, 14, 1, 1, 42, 462, 462, 42, 1, 1, 132, 6006, 24024, 6006, 132, 1, 1, 429, 87516, 1662804, 1662804, 87516, 429, 1, 1, 1430, 1385670, 140229804, 701149020, 140229804, 1385670, 1430, 1, 1, 4862, 23371634, 13672405890, 396499770810, 396499770810, 13672405890, 23371634, 4862, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| Multidimensional Catalan numbers; a special case of the "hook-number formula".
Number of paths from (0,0,...,0) to (n,n,...n) in m dimensions, all coordinates increasing: if (x_1,x_2,..x_m) is on the path, then x_1 <= x_2 <= .. <= x_m. Number of ways to label an n by m array with all the values 1..n*m such that each row and column is strictly increasing. Number of rectangular Young Tableaux. Number of linear extensions of the n X m lattice (the divisor lattice of a number having exactly two prime divisors). - Mitch Harris (Harris.Mitchell (AT) mgh.harvard.edu), Dec 27, 2005
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REFERENCES
| J. S. Frame, G. de B. Robinson and R. M. Thrall, The hook graphs of a symmetric group, Canad. J. Math. 6 (1954), pp. 316-.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 7.23.19(b).
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FORMULA
| T(m, n) = 0!*1!*..*(n-1)! *(m*n)! / ( m!*(m+1)!*..*(m+n-1)! ).
T(m, n) = A000142(mn)*A000178(m-1)*A000178(n-1)/A000178(m+n-1) = A000142(A004247(m, n)) * A007318(m+n, n)/A009963(m+n, n) - Henry Bottomley (se16(AT)btinternet.com), May 22 2002
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EXAMPLE
| Array begins:
1 1 1 1 1 1 ...
1 2 5 14 42 132 ...
1 5 42 462 6006 87516 ...
1 14 462 24024 1662804 140229804 ...
...
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MAPLE
| T:= (m, n)-> mul (i!, i=0..n-1) *(m*n)! /mul ((m+i)!, i=0..n-1):
seq (seq (T(n, d-n), n=1..d-1), d=2..11);
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MATHEMATICA
| maxm = 10; t[m_, n_] := Product[k!, {k, 0, n - 1}]*(m*n)! / Product[k!, {k, m, m + n - 1}]; Flatten[ Table[t[m + 1 - n, n], {m, 1, maxm}, {n, 1, m}]] (* From Jean-François Alcover, Sep 21 2011 *)
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PROG
| (PARI) A(i, j)=if(i<0|j<0, 0, (i*j)!/prod(k=1, i+j-1, k^vecmin([k, i, j, i+j-k]))) - Michael Somos, Jan 28 2004
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CROSSREFS
| Rows give A000108 (Catalan numbers), A005789, A005790, A005791. Diagonals give A039622, A060855, A060856.
Sequence in context: A099927 A187617 A128612 * A091378 A156045 A119687
Adjacent sequences: A060851 A060852 A060853 * A060855 A060856 A060857
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KEYWORD
| nonn,tabl,easy,nice
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AUTHOR
| R. H. Hardin (rhhardin(AT)att.net), May 03 2001
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EXTENSIONS
| More terms from Frank.Ellermann(AT)t-online.de, May 21 2001
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