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%I #102 Apr 20 2023 17:27:44
%S 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 6006,132,1,1,429,87516,1662804,1662804,87516,429,1,1,1430,1385670,
%U 140229804,701149020,140229804,1385670,1430,1,1,4862,23371634,13672405890,396499770810,396499770810,13672405890,23371634,4862,1
%N 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 Multidimensional Catalan numbers; a special case of the "hook-number formula".
%C 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
%C 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
%D 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).
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 7.23.19(b).
%H Alois P. Heinz, <a href="/A060854/b060854.txt">Antidiagonals n = 1..36</a>
%H Albrecht Böttcher, <a href="https://doi.org/10.1002/mana.19891440105">Wiener-Hopf Determinants with Rational Symbols</a>, Math. Nachr. 144 (1989), 39-64.
%H Freddy Cachazo and Nick Early, <a href="https://arxiv.org/abs/2003.07958">Minimal Kinematics: An all k and n peek into Trop^+G(k,n)</a>, arXiv:2003.07958 [hep-th], 2020.
%H Freddy Cachazo and Nick Early, <a href="https://arxiv.org/abs/2010.09708">Planar Kinematics: Cyclic Fixed Points, Mirror Superpotential, k-Dimensional Catalan Numbers, and Root Polytopes</a>, arXiv:2010.09708 [math.CO], 2020.
%H Freddy Cachazo and Nick Early, <a href="https://arxiv.org/abs/2204.01743">Biadjoint Scalars and Associahedra from Residues of Generalized Amplitudes</a>, arXiv:2204.01743 [hep-th], 2022.
%H Nick Early, <a href="https://arxiv.org/abs/2106.07142">Planarity in Generalized Scattering Amplitudes: PK Polytope, Generalized Root Systems and Worldsheet Associahedra</a>, arXiv:2106.07142 [math.CO], 2021, see p. 14.
%H Ömer Eğecioğlu, <a href="https://ajc.maths.uq.edu.au/pdf/43/ajc_v43_p307.pdf">On Böttcher's mysterious identity</a>, Australasian Journal of Combinatorics, Volume 43 (2009), 307-316.
%H Paul Drube, <a href="http://arxiv.org/abs/1606.04869">Generating Functions for Inverted Semistandard Young Tableaux and Generalized Ballot Numbers</a>, arXiv:1606.04869 [math.CO], 2016.
%H Claudio Fontanari, <a href="https://arxiv.org/abs/2206.06709">Guido Castelnuovo and his heritage: geometry, combinatorics, teaching</a>, arXiv:2206.06709 [math.HO], 2022. See pp. 2-3.
%H J. S. Frame, G. de B. Robinson and R. M. Thrall, <a href="http://dx.doi.org/10.4153/CJM-1954-030-1">The hook graphs of a symmetric group</a>, Canad. J. Math. 6 (1954), pp. 316-324.
%H Alexander Garver and Thomas McConville, <a href="https://doi.org/10.5802/alco.142">Chapoton triangles for nonkissing complexes</a>, Algebraic Combinatorics, 3 (2020), pp. 1331-1363.
%H K. Gorska and K. A. Penson, <a href="http://arxiv.org/abs/1304.6008">Multidimensional Catalan and related numbers as Hausdorff moments</a>, arXiv preprint arXiv:1304.6008 [math.CO], 2013.
%H F. Santos, C. Stump, and V. Welker, <a href="https://hal.inria.fr/hal-01207539">Noncrossing sets and a Graßmannian associahedron</a>, in FPSAC 2014, Chicago, USA; Discrete Mathematics and Theoretical Computer Science (DMTCS) Proceedings, 2014, 609-620.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hook_length_formula">Hook length formula</a>
%F T(m, n) = 0!*1!*..*(n-1)! *(m*n)! / ( m!*(m+1)!*..*(m+n-1)! ).
%F 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
%e Array begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 2, 5, 14, 42, 132, ...
%e 1, 5, 42, 462, 6006, 87516, ...
%e 1, 14, 462, 24024, 1662804, 140229804, ...
%e 1, 42, 6006, 1662804, 701149020, 396499770810, ...
%e 1, 132, 87516, 140229804, 396499770810, 1671643033734960, ...
%p T:= (m, n)-> (m*n)! * mul(i!/(m+i)!, i=0..n-1):
%p seq(seq(T(n, 1+d-n), n=1..d), d=1..10);
%t 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 *)
%t 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 *)
%o (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 Rows give A000108 (Catalan numbers), A005789, A005790, A005791, A321975, A321976, A321977, A321978.
%Y Diagonals give A039622, A060855, A060856.
%Y Cf. A227578. - _Alois P. Heinz_, Jul 18 2013
%Y Cf. A321716.
%K nonn,tabl,easy,nice
%O 1,5
%A _R. H. Hardin_, May 03 2001
%E More terms from _Frank Ellermann_, May 21 2001
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