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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. 33
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; text; internal format)
OFFSET

1,5

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, Dec 27 2005

REFERENCES

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 7.23.19(b).

LINKS

Alois P. Heinz, Antidiagonals n = 1..36

Paul Drube, Generating Functions for Inverted Semistandard Young Tableaux and Generalized Ballot Numbers, arXiv:1606.04869 [math.CO], 2016.

J. S. Frame, G. de B. Robinson and R. M. Thrall, The hook graphs of a symmetric group, Canad. J. Math. 6 (1954), pp. 316-324.

K. Gorska and K. A. Penson, Multidimensional Catalan and related numbers as Hausdorff moments, arXiv preprint arXiv:1304.6008 [math.CO], 2013.

F. Santos, C. Stump, V. Welker, Noncrossing sets and a Graßmannian associahedron, in FPSAC 2014, Chicago, USA; Discrete Mathematics and Theoretical Computer Science (DMTCS) Proceedings, 2014, 609-620.

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, May 22 2002

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, ...

1,  42,  6006,   1662804,    701149020,     396499770810, ...

1, 132, 87516, 140229804, 396499770810, 1671643033734960, ...

MAPLE

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);

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}]] (* 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 *)

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 */

CROSSREFS

Rows give A000108 (Catalan numbers), A005789, A005790, A005791. Diagonals give A039622, A060855, A060856.

Cf. A227578. - Alois P. Heinz, Jul 18 2013

Sequence in context: A128612 A284731 A211400 * A091378 A156045 A119687

Adjacent sequences:  A060851 A060852 A060853 * A060855 A060856 A060857

KEYWORD

nonn,tabl,easy,nice

AUTHOR

R. H. Hardin, May 03 2001

EXTENSIONS

More terms from Frank Ellermann, May 21 2001

STATUS

approved

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Last modified December 12 20:04 EST 2017. Contains 295954 sequences.