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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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38
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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
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OFFSET
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1,5
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COMMENTS
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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, Dec 27 2005
Given m*n lines in a {(m + 1)(n - 1)}-dimensional space, T(m, n) is the number of {n*(m-1)-1}-dimensional spaces cutting these lines in points (see Fontanari and Castelnuovo). - Stefano Spezia, Jun 19 2022
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REFERENCES
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Guido Castelnuovo, Numero degli spazi che segano più rette in uno spazio ad n dimensioni, Rendiconti della R. Accademia dei Lincei, s. IV, vol. V, 4 agosto 1889. In Guido Castelnuovo, Memorie scelte, Zanichelli, Bologna 1937, pp. 55-64 (in Italian).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 7.23.19(b).
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LINKS
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FORMULA
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T(m, n) = 0!*1!*..*(n-1)! *(m*n)! / ( m!*(m+1)!*..*(m+n-1)! ).
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EXAMPLE
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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, ...
1, 42, 6006, 1662804, 701149020, 396499770810, ...
1, 132, 87516, 140229804, 396499770810, 1671643033734960, ...
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MAPLE
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T:= (m, n)-> (m*n)! * mul(i!/(m+i)!, i=0..n-1):
seq(seq(T(n, 1+d-n), n=1..d), d=1..10);
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MATHEMATICA
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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}]] (* Jean-François Alcover, Sep 21 2011 *)
Table[ BarnesG[n+1]*(n*(m-n+1))!*BarnesG[m-n+2] / BarnesG[m+2], {m, 1, 10}, {n, 1, m}] // Flatten (* Jean-François Alcover, Jan 30 2016 *)
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PROG
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(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
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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