%I A060854
%S A060854 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,
%T A060854 6006,132,1,1,429,87516,1662804,1662804,87516,429,1,1,1430,1385670,
%U A060854 140229804,701149020,140229804,1385670,1430,1,1,4862,23371634
%N A060854 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.
%C A060854 Multidimensional Catalan numbers; a special case of the "hook-number 
               formula".
%C A060854 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
%D A060854 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-.
%D A060854 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see 
               Example 7.23.19(b).
%F A060854 T(m, n) = 0!*1!*..*(n-1)! *(m*n)! / ( m!*(m+1)!*..*(m+n-1)! )
%F A060854 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
%e A060854 Array begins
%e A060854 1 1 1 1 1 1 1 ...
%e A060854 1 2 5 14 42 132 ...
%e A060854 1 5 42 462 6006 ...
%e A060854 1 14 462 24024 ...
%e A060854 ...
%o A060854 (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
%Y A060854 Rows give A000108 (Catalan numbers), A005789, A005790, A005791. Diagonals 
               give A039622, A060855, A060856.
%Y A060854 Sequence in context: A139332 A099927 A128612 this_sequence A091378 A156045 
               A119687
%Y A060854 Adjacent sequences: A060851 A060852 A060853 this_sequence A060855 A060856 
               A060857
%K A060854 nonn,tabl,easy,nice
%O A060854 1,5
%A A060854 R. H. Hardin (rhhardin(AT)att.net), May 03 2001
%E A060854 More terms from Frank.Ellermann(AT)t-online.de, May 21 2001