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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. 38
 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 Albrecht Böttcher, Wiener-Hopf Determinants with Rational Symbols, Math. Nachr. 144 (1989), 39-64. Freddy Cachazo and Nick Early, Minimal Kinematics: An all k and n peek into Trop^+G(k,n), arXiv:2003.07958 [hep-th], 2020. Freddy Cachazo and Nick Early, Planar Kinematics: Cyclic Fixed Points, Mirror Superpotential, k-Dimensional Catalan Numbers, and Root Polytopes, arXiv:2010.09708 [math.CO], 2020. Nick Early, Planarity in Generalized Scattering Amplitudes: PK Polytope, Generalized Root Systems and Worldsheet Associahedra, arXiv:2106.07142 [math.CO], 2021, see p. 14. Ömer Eğecioğlu, On Böttcher's mysterious identity, Australasian Journal of Combinatorics, Volume 43 (2009), Pages 307-316. 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. Alexander Garver and Thomas McConville, Chapoton triangles for nonkissing complexes, University of Michigan (2020). 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, and V. Welker, Noncrossing sets and a Graßmannian associahedron, in FPSAC 2014, Chicago, USA; Discrete Mathematics and Theoretical Computer Science (DMTCS) Proceedings, 2014, 609-620. Wikipedia, Hook length formula FORMULA T(m, n) = 0!*1!*..*(n-1)! *(m*n)! / ( m!*(m+1)!*..*(m+n-1)! ). T(m, n) = A000142(m*n)*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, A321975, A321976, A321977, A321978. Diagonals give A039622, A060855, A060856. Cf. A227578. - Alois P. Heinz, Jul 18 2013 Cf. A321716. 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 January 27 06:18 EST 2022. Contains 350601 sequences. (Running on oeis4.)