A005790 -id:A005790 - cp's OEIS frontend

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.

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

Offset: 1

R. H. Hardin, May 03 2001

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
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

			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, ...

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

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.
Freddy Cachazo and Nick Early, Biadjoint Scalars and Associahedra from Residues of Generalized Amplitudes, arXiv:2204.01743 [hep-th], 2022.
Andrzej Dudek, Jarosław Grytczuk, Jakub Przybyło, and Andrzej Ruciński, Homogeneous substructures in random ordered hyper-matchings, arXiv:2507.20374 [math.CO], 2025. See p. 19.
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), 307-316.
Paul Drube, Generating Functions for Inverted Semistandard Young Tableaux and Generalized Ballot Numbers, arXiv:1606.04869 [math.CO], 2016.
Claudio Fontanari, Guido Castelnuovo and his heritage: geometry, combinatorics, teaching, arXiv:2206.06709 [math.HO], 2022. See pp. 2-3.
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, Algebraic Combinatorics, 3 (2020), pp. 1331-1363.
Katarzyna Górska and Karol A. Penson, Multidimensional Catalan and related numbers as Hausdorff moments, arXiv preprint arXiv:1304.6008 [math.CO], 2013.
Owen John Levens, Joel Brewster Lewis, and Bridget Eileen Tenner, Global patterns in signed permutations, arXiv:2504.13108 [math.CO], 2025. See p. 18.
Dimana Miroslavova Pramatarova, Investigating the Periodicity of Weighted Catalan Numbers and Generalizing Them to Higher Dimensions, MIT Res. Sci. Instit. (2025). See p. 5.
Francisco Santos, Christian Stump, and Volkmar 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

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.

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);

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 *)

{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 */

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

More terms from Frank Ellermann, May 21 2001

A039622 Number of n X n Young tableaux.

Original entry on oeis.org

1, 1, 2, 42, 24024, 701149020, 1671643033734960, 475073684264389879228560, 22081374992701950398847674830857600, 220381378415074546123953914908618547085974856000, 599868742615440724911356453304513631101279740967209774643120000

Offset: 0

Floor van Lamoen

Number of arrangements of 1,2,...,n^2 in an n X n array such that each row and each column is increasing. The problem for a 5 X 5 array was recently posed and solved in the College Mathematics Journal. See the links.
This is the factor g_n that appears in a conjectured formula for 2n-th moment of the Riemann zeta function on the critical line. (See Conrey articles.) - Michael Somos, Apr 15 2003 [Comment revised by N. J. A. Sloane, Jun 21 2016]
Number of linear extensions of the n X n lattice. - Mitch Harris, Dec 27 2005

			Using the hook length formula, a(4) = (16)!/(7*6^2*5^3*4^4*3^3*2^2) = 24024.

M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 284.

Alois P. Heinz, Table of n, a(n) for n = 0..30
P. Aluffi, Degrees of projections of rank loci, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."]
Joerg Arndt, The a(3)=42 3 X 3 Young tableaux
J. B. Conrey, The Riemann Hypothesis, Notices Amer. Math. Soc., 50 (No. 3, March 2003), 341-353. See p. 349.
J. B. Conrey, Review of H. Iwaniec, "Lectures on the Riemann Zeta Function" (AMS, 2014), Bull. Amer. Math. Soc., 53 (No. 3, 2016), 507-512.
P.-O. Dehaye, Combinatorics of the lower order terms in the moment conjectures: the Riemann zeta function, arXiv preprint arXiv:1201.4478 [math.NT], 2012.
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.
Curtis Greene and Brady Haran, Shapes and Hook Numbers, Numberphile video (2016)
Curtis Greene and Brady Haran, Shapes and Hook Numbers (extra footage) (2016)
Zachary Hamaker and Eric Marberg, Atoms for signed permutations, arXiv:1802.09805 [math.CO], 2018.
Alejandro H. Morales, I. Pak, and G. Panova, Why is pi < 2 phi?, Preprint, 2016; The American Mathematical Monthly, Volume 125, 2018 - Issue 8.
Alan H. Rapoport (proposer), Solution to Problem 639: A Square Young Tableau, College Mathematics Journal, Vol. 30 (1999), no. 5, pp. 410-411.
Index entries for sequences related to Young tableaux.

Main diagonal of A060854.
Also a(2)=A000108(2), a(3)=A005789(3), a(4)=A005790(4), a(5)=A005791(5).
Cf. A000178, A000984, A079478, A088020, A107252, A107254, A127223, A153452.

A039622:= func< n | n eq 0 select 1 else Factorial(n^2)*(&*[Factorial(j)/Factorial(n+j): j in [0..n-1]]) >;
[A039622(n): n in [0..12]]; // G. C. Greubel, Apr 21 2021

a:= n-> (n^2)! *mul(k!/(n+k)!, k=0..n-1):
seq(a(n), n=0..12);  # Alois P. Heinz, Apr 10 2012

a[n_]:= (n^2)!*Product[ k!/(n+k)!, {k, 0, n-1}]; Table[ a[n], {n, 0, 12}] (* Jean-François Alcover, Dec 06 2011, after Pari *)

PARI
```
a(n)=(n^2)!*prod(k=0,n-1,k!/(n+k)!)
```

def A039622(n): return factorial(n^2)*product( factorial(j)/factorial(n+j) for j in (0..n-1))
[A039622(n) for n in (0..12)] # G. C. Greubel, Apr 21 2021

a(n) = (n^2)! / Product_{k=1..2n-1} k^(n - |n-k|).
a(n) = 0!*1!*...*(k-1)! *(k*n)! / ( n!*(n+1)!*...*(n+k-1)! ) for k=n.
a(n) = A088020(n)/A107254(n) = A088020(n)*A000984(n)/A079478(n). - Henry Bottomley, May 14 2005
a(n) = A153452(prime(n)^n). - Naohiro Nomoto, Jan 01 2009
a(n) ~ sqrt(Pi) * n^(n^2+11/12) * exp(n^2/2+1/12) / (A * 2^(2*n^2-7/12)), where A = 1.28242712910062263687534256886979... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Feb 10 2015
From Peter Luschny, May 20 2019: (Start)
a(n) = (G(1+n)*G(2+n)^(2-n)*(n^2)!*(G(3+n)/Gamma(2+n))^(n-1))/(G(1+2*n)*n!) where G(x) is the Barnes G function.
a(n) = A127223(n) / A107252(n). (End)
a(n) = (Gamma(n^2 +1)/Gamma(n+1))*(G(n+1)*G(n+2)/G(2*n+1)), where G(n) is the Barnes G-function. - G. C. Greubel, Apr 21 2021
a(n+2) = (n+2) * A060856(n+1) for n >= 0. - Tom Copeland, May 30 2022

A245173 Triangle read by rows: coefficients of the polynomials A_{3,4}(n,k).

Original entry on oeis.org

1, 0, 1, 0, 1, 6, 6, 1, 0, 1, 22, 113, 190, 113, 22, 1, 0, 1, 53, 710, 3548, 7700, 7700, 3548, 710, 53, 1, 0, 1, 105, 2856, 30422, 151389, 385029, 523200, 385029, 151389, 30422, 2856, 105, 1

Offset: 0

N. J. A. Sloane, Jul 13 2014

From Per W. Alexandersson, Sep 05 2019: (Start)
Let F(n,0) = 1/(1-z), and F(n,k) = z^(n-1)*( d^n/dz^n F(n,k-1) ).
The n-th row is then given by the coefficients of the monic polynomial factor in the numerator of F(n,4).
The (k+1)-th entry in row n is given by the number of standard Young tableaux of rectangular shape (n,n,n,n), with exactly k descents. (Proved by G. Panova on MathOverflow, see Links.) (End)

			Triangle begins:
  1;
  0, 1;
  0, 1, 6, 6, 1;
  0, 1, 22, 113, 190, 113, 22, 1;
  0, 1, 53, 710, 3548, 7700, 7700, 3548, 710, 53, 1;
  0, 1, 105, 2856, 30422, 151389, 385029, 523200, 385029, 151389, 30422, 2856, 105, 1;
...

Per W. Alexandersson, Table of n, a(n) for n = 0..1365
J. Agapito, On symmetric polynomials with only real zeros and nonnegative gamma-vectors, Linear Algebra and its Applications, Volume 451, 15 June 2014, Pages 260-289.
Greta Panova, Iterated derivative and rectangular standard Young tableaux, version: 2019-09-05.

Cf. A245167, A245168, A245169, A245170, A245171, A245172.

Row sums are given by A005790.

GG[a_, b_] := z (Product[(k)!/(a + k)!, {k, 0, b - 1}]) z^(1 - a) (1 - z)^(a b + 1) Nest[Simplify[z^(a - 1) D[#, {z, a}]] &, 1/(1 - z), b];
Table[CoefficientList[GG[a, 4] // Together, z], {a, 1, 8}] (* Per W. Alexandersson, Sep 05 2019 *)

A215204 Number A(n,k) of solid standard Young tableaux of cylindrical shape lambda X k, where lambda ranges over all partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 4, 4, 5, 1, 1, 10, 26, 10, 7, 1, 1, 28, 276, 258, 26, 11, 1, 1, 84, 3740, 14318, 3346, 76, 15, 1, 1, 264, 58604, 1161678, 1214358, 54108, 232, 22, 1, 1, 858, 1010616, 118316062, 741215012, 150910592, 1054256, 764, 30

Offset: 0

Alois P. Heinz, Aug 05 2012

			Square array A(n,k) begins:
:  1,  1,     1,         1,            1,                1, ...
:  1,  1,     1,         1,            1,                1, ...
:  2,  2,     4,        10,           28,               84, ...
:  3,  4,    26,       276,         3740,            58604, ...
:  5, 10,   258,     14318,      1161678,        118316062, ...
:  7, 26,  3346,   1214358,    741215012,     620383261034, ...
: 11, 76, 54108, 150910592, 840790914296, 7137345113624878, ...

Alois P. Heinz, Antidiagonals n = 0..15, flattened
S. B. Ekhad and D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Columns k=0-5 give: A000041, A000085, A215266, A290202, A290214, A290274.
Rows n=0+1, 2-5 give: A000012, 2*A000108, 2*A005789 + A006335, 2*A005790 + 2*A213978 + A114714, 2*A005791 + 2*A215220 + 2*A213932 + A214638.
Main diagonal gives A290225.

b:= proc(l) option remember; local m; m:= nops(l);
      `if`({map(x-> x[], l)[]}minus{0}={}, 1, add(add(`if`(l[i][j]>
      `if`(i=m or nops(l[i+1])
      `if`(nops(l[i])=j, 0, l[i][j+1]), b(subsop(i=subsop(
       j=l[i][j]-1, l[i]), l)), 0), j=1..nops(l[i])), i=1..m))
    end:
g:= proc(n, i, k, l) `if`(n=0 or i=1, b(map(x-> [k$x], [l[], 1$n])),
       add(g(n-i*j, i-1, k, [l[], i$j]), j=0..n/i))
    end:
A:= (n, k)-> g(n, n, k, []):
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[l_] := b[l] = With[{m = Length[l]}, If[Union[l // Flatten] ~Complement~ {0} == {}, 1, Sum[Sum[If[l[[i, j]] > If[i == m || Length[l[[i + 1]]] < j, 0, l[[i + 1, j]]] && l[[i, j]] > If[Length[l[[i]]] == j, 0, l[[i, j + 1]]], b[ReplacePart[l, i -> ReplacePart[l[[i]], j -> l[[i, j]] - 1]]], 0], {j, 1, Length[l[[i]]]}], {i, 1, m}]]];
g[n_, i_, k_, l_] := If[n == 0 || i == 1, b[Table[k, {#}]& /@ Join[l, Table[1, {n}]]], Sum[g[n - i*j, i - 1, k, Join[l, Table[i, {j}]]], {j, 0, n/i}]];
A[n_, k_] := g[n, n, k, {}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Sep 24 2022, after Alois P. Heinz *)

A321975 6-dimensional Catalan numbers.

Original entry on oeis.org

1, 1, 132, 87516, 140229804, 396499770810, 1671643033734960, 9490348077234178440, 67867669180627125604080, 583692803893929928888544400, 5838544419011620940996212276800, 66244124978105851196543024492572800, 836288764382254532915188713779640302400, 11570895443447601081407359451642915869302000

Offset: 0

Seiichi Manyama, Nov 23 2018

nonn

Number of n X 6 Young tableaux.

Seiichi Manyama, Table of n, a(n) for n = 0..222
Wikipedia, Hook length formula

Row 6 of A060854.

Cf. A000108, A005789, A005790, A005791.

List([0..15],n->34560*Factorial(6*n)/Product([0..5],k->Factorial(n+k))); # Muniru A Asiru, Nov 25 2018

[34560*Factorial(6*n)/(Factorial(n)*Factorial(n + 1)*Factorial(n + 2)*Factorial(n + 3)*Factorial(n + 4)*Factorial(n + 5)): n in [0..15]]; // Vincenzo Librandi, Nov 24 2018

a:= n-> (6*n)! * mul(i!/(6+i)!, i=0..n-1):
seq(a(n), n=0..14);  # Alois P. Heinz, Nov 25 2018

Table[34560 (6 n)! / (n! (n + 1)! (n + 2)! (n + 3)! (n + 4)! (n + 5)!), {n, 0, 60}] (* Vincenzo Librandi, Nov 24 2018 *)

{a(n) = 34560*(6*n)!/(n!*(n+1)!*(n+2)!*(n+3)!*(n+4)!*(n+5)!)}

a(n) = 0!*1!*...*5! * (6*n)! / ( n!*(n+1)!*...*(n+5)! ).

a(n) ~ 5 * 2^(6*n + 6) * 3^(6*n + 7/2) / (Pi^(5/2) * n^(35/2)). - Vaclav Kotesovec, Nov 23 2018

A321976 7-dimensional Catalan numbers.

Original entry on oeis.org

1, 1, 429, 1385670, 13672405890, 278607172289160, 9490348077234178440, 475073684264389879228560, 32103104214166146088869942000, 2760171874087743799855959353857200, 289232890341906497299306268771988273600, 35764585916110766978895474668714467232388000

Offset: 0

Seiichi Manyama, Nov 23 2018

nonn

Number of n X 7 Young tableaux.

Seiichi Manyama, Table of n, a(n) for n = 0..177
Wikipedia, Hook length formula

Row 7 of A060854.

Cf. A000108, A005789, A005790, A005791, A321975.

List([0..15],n->24883200*Factorial(7*n)/Product([0..6],k->Factorial(n+k))); # Muniru A Asiru, Nov 25 2018

[24883200*Factorial(7*n) / (Factorial(n)*Factorial(n + 1)*Factorial(n + 2)*Factorial(n + 3)*Factorial(n + 4)*Factorial(n + 5)*Factorial(n + 6)): n in [0..15]]; // Vincenzo Librandi, Nov 24 2018

Table[24883200*(7*n)!/(n!*(n+1)!*(n+2)!*(n+3)!*(n+4)!*(n+5)!*(n+6)!),{n,0,15}] (* Vincenzo Librandi, Nov 24 2018 *)

{a(n) = 24883200*(7*n)!/(n!*(n+1)!*(n+2)!*(n+3)!*(n+4)!*(n+5)!*(n+6)!)}

a(n) = 0!*1!*...*6! * (7*n)! / ( n!*(n+1)!*...*(n+6)! ).

a(n) ~ 3110400 * 7^(7*n + 1/2) / (Pi^3 * n^24). - Vaclav Kotesovec, Nov 23 2018

A321977 8-dimensional Catalan numbers.

Original entry on oeis.org

1, 1, 1430, 23371634, 1489877926680, 231471904322784840, 67867669180627125604080, 32103104214166146088869942000, 22081374992701950398847674830857600, 20535535214275361308250745082811167425600, 24486819823897171791550434989846505231774984000

Offset: 0

Seiichi Manyama, Nov 23 2018

nonn

Number of n X 8 Young tableaux.

Seiichi Manyama, Table of n, a(n) for n = 0..146
Wikipedia, Hook length formula

Row 8 of A060854.

Cf. A000108, A005789, A005790, A005791, A321975, A321976.

List([0..15],n->125411328000*Factorial(8*n)/Product([0..7],k->Factorial(n+k))); # Muniru A Asiru, Nov 25 2018

[125411328000*Factorial(8*n)/(Factorial(n)*Factorial(n + 1)*Factorial(n + 2)*Factorial(n + 3)*Factorial(n + 4)*Factorial(n + 5)*Factorial(n + 6)*Factorial(n + 7)): n in [0..15]]; // Vincenzo Librandi, Nov 24 2018

Table[125411328000 (8 n)! / (n! (n+1)! (n+2)! (n+3)! (n+4)! (n+5)! (n+6)! (n + 7)!), {n, 0, 15}] (* Vincenzo Librandi, Nov 24 2018 *)

a(n) = 0!*1!*...*7! * (8*n)! / ( n!*(n+1)!*...*(n+7)! ).

a(n) ~ 1913625 * 2^(24*n + 14) / (Pi^(7/2) * n^(63/2)). - Vaclav Kotesovec, Nov 23 2018

A321716 Triangle read by rows: T(n,k) is the number of n X k Young tableaux, where 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 5, 42, 1, 1, 14, 462, 24024, 1, 1, 42, 6006, 1662804, 701149020, 1, 1, 132, 87516, 140229804, 396499770810, 1671643033734960, 1, 1, 429, 1385670, 13672405890, 278607172289160, 9490348077234178440, 475073684264389879228560

Offset: 0

Seiichi Manyama, Nov 17 2018

			T(4,3) = 12! / ((6*5*4)*(5*4*3)*(4*3*2)*(3*2*1)) = 462.
Triangle begins:
  1;
  1, 1;
  1, 1,   2;
  1, 1,   5,    42;
  1, 1,  14,   462,     24024;
  1, 1,  42,  6006,   1662804,    701149020;
  1, 1, 132, 87516, 140229804, 396499770810, 1671643033734960;

Seiichi Manyama, Rows n = 0..30, flattened
Wikipedia, Hook length formula
Index entries for sequences related to Young tableaux.

Cf. A000178, A005789, A005790, A005791, A039622, A060854

A321716:= func< n,k | n eq 0 select 1 else Factorial(n*k)/(&*[ Round(Gamma(j+k)/Gamma(j)): j in [1..n]]) >;
[A321716(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 04 2021

T[n_, k_]:= (n*k)!/Product[Product[i+j-1, {j,1,k}], {i,1,n}]; Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Amiram Eldar, Nov 17 2018 *)
T[n_, k_]:= (n*k)!*BarnesG[n+1]*BarnesG[k+1]/BarnesG[n+k+1];
Table[T[n, k], {n, 0, 5}, {k, 0, n}] //Flatten (* G. C. Greubel, May 04 2021 *)

def A321716(n,k): return factorial(n*k)/product( gamma(j+k)/gamma(j) for j in (1..n) )
flatten([[A321716(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 04 2021

T(n, k) = (n*k)! / (Product_{i=1..n} Product_{j=1..k} (i+j-1)).
T(n, k) = A060854(n,k) for n,k > 0.
T(n, n) = A039622(n).
T(n, k) = (n*k)!*BarnesG(n+1)*BarnesG(k+1)/BarnesG(n+k+1), where BarnesG(n) = A000178. - G. C. Greubel, May 04 2021

A321978 9-dimensional Catalan numbers.

Original entry on oeis.org

1, 1, 4862, 414315330, 177295473274920, 219738059326729823880, 583692803893929928888544400, 2760171874087743799855959353857200, 20535535214275361308250745082811167425600, 220381378415074546123953914908618547085974856000

Offset: 0

Seiichi Manyama, Nov 23 2018

nonn

Number of n X 9 Young tableaux.

Seiichi Manyama, Table of n, a(n) for n = 0..124
Wikipedia, Hook length formula

Row 9 of A060854.

Cf. A000108, A005789, A005790, A005791, A321975, A321976, A321977.

List([0..10],n->5056584744960000*Factorial(9*n)/Product([0..8],k->Factorial(n+k))); # Muniru A Asiru, Nov 25 2018

[5056584744960000*Factorial(9*n)/(Factorial(n)*Factorial(n + 1)*Factorial(n + 2)*Factorial(n + 3)*Factorial(n + 4)*Factorial(n + 5)*Factorial(n + 6)*Factorial(n + 7)*Factorial(n + 8)): n in [0..15]]; // Vincenzo Librandi, Nov 24 2018

Table[5056584744960000 (9 n)! / (n! (n + 1)! (n + 2)! (n + 3)! (n + 4)! (n + 5)! (n + 6)! (n + 7)! (n + 8)!), {n, 0, 15}] (* Vincenzo Librandi, Nov 24 2018 *)

a(n) = 0!*1!*...*8! * (9*n)! / ( n!*(n+1)!*...*(n+8)! ).

a(n) ~ 16056320000 * 3^(18*n + 10) / (Pi^4 * n^40). - Vaclav Kotesovec, Nov 23 2018

cp's OEIS Frontend

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.

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A039622 Number of n X n Young tableaux.

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A245173 Triangle read by rows: coefficients of the polynomials A_{3,4}(n,k).

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A215204 Number A(n,k) of solid standard Young tableaux of cylindrical shape lambda X k, where lambda ranges over all partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A321975 6-dimensional Catalan numbers.

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A321976 7-dimensional Catalan numbers.

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A321977 8-dimensional Catalan numbers.