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Search: id:A060854
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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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13
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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
(list; table; graph; listen)
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OFFSET
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1,5
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COMMENT
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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 (Harris.Mitchell (AT) mgh.harvard.edu), Dec 27, 2005
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REFERENCES
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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
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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
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Array begins
1 1 1 1 1 1 1 ...
1 2 5 14 42 132 ...
1 5 42 462 6006 ...
1 14 462 24024 ...
...
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PROGRAM
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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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Rows give A000108 (Catalan numbers), A005789, A005790, A005791. Diagonals give A039622, A060855, A060856.
Sequence in context: A139332 A099927 A128612 this_sequence A091378 A119687 A086856
Adjacent sequences: A060851 A060852 A060853 this_sequence A060855 A060856 A060857
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KEYWORD
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nonn,tabl,easy,nice
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AUTHOR
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R. H. Hardin (rhh(AT)cadence.com), May 03 2001
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EXTENSIONS
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More terms from Frank.Ellermann(AT)t-online.de, May 21 2001
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