A005789 -id:A005789 - cp's OEIS frontend

A151334 Duplicate of A005789.

1, 1, 5, 42, 462, 6006, 87516, 1385670, 23371634, 414315330, 7646001090, 145862174640, 2861142656400, 57468093927120

Offset: 0

dead

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

A005790 4-dimensional Catalan numbers.

Original entry on oeis.org

1, 1, 14, 462, 24024, 1662804, 140229804, 13672405890, 1489877926680, 177295473274920, 22661585038594320, 3073259571003214320, 438091463242348309440, 65166105157299311029200, 10056663345892631910888600, 1602608179958939072505281850, 262708662267696303439658400600

Offset: 0

N. J. A. Sloane

Number of standard tableaux of shape (n,n,n,n). - Emeric Deutsch, May 13 2004

The prime terms (as defined in A268538) are 1, 1, 10, 320, 16764, 1171355, 99315236, 9691755128, 1053114415100, ... - R. J. Mathar, Feb 27 2018

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Snover, Stephen L.; Troyer, Stephanie F.; A four-dimensional Catalan formula. Proceedings of the Nineteenth Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, MB, 1989). Congr. Numer. 75 (1990), 123-126.

Seiichi Manyama, Table of n, a(n) for n = 0..423 (terms 1..130 from Alois P. Heinz)
Shalosh B. Ekhad and Doron Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux. Also arXiv preprint arXiv:1202.6229, 2012. - _N. J. A. Sloane_, Jul 07 2012
Michaël Moortgat, The Tamari order for D^3 and derivability in semi-associative Lambek-Grishin Calculus, 15th Workshop: Computational Logic and Applications (CLA 2020).
Katarzyna Górska and Karol A. Penson, Multidimensional Catalan and related numbers as Hausdorff moments, arXiv preprint arXiv:1304.6008 [math.CO], 2013.
Dimana Miroslavova Pramatarova, Investigating the Periodicity of Weighted Catalan Numbers and Generalizing Them to Higher Dimensions, MIT Res. Sci. Instit. (2025). See p. 13.
Stephen Snover, Letter to N. J. A. Sloane, May 1991
Stephanie F. Troyer and Stephen L. Snover, m-Dimensional Catalan numbers, Preprint, 1989. (Annotated scanned copy)

A row of A060854.

Cf. A000108 (Catalan numbers), A005789, A005791.

[12*Factorial(4*n)/(Factorial(n)*Factorial(n+1)*Factorial(n+2) *Factorial(n+3)): n in [0..20]]; // Vincenzo Librandi, Nov 23 2018

a:= n-> (4*n)! * mul(i!/(4+i)!, i=0..n-1):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 25 2012

Table[12*(4*n)!/(n!*(n+1)!*(n+2)!*(n+3)!), {n, 0, 20}] (* Vaclav Kotesovec, Nov 18 2016 *)

vector(20, n, n--; 12*(4*n)!/(n!*(n+1)!*(n+2)!*(n+3)!)) \\ G. C. Greubel, Nov 23 2018

[12*factorial(4*n)/(factorial(n)*factorial(n+1)*factorial(n+2) *factorial(n+3)) for n in range(20)] # G. C. Greubel, Nov 23 2018

a(n) = 12*(4*n)!/(n! *(n+1)! *(n+2)! *(n+3)!).
G.f.: 4_F_3 ( [ 1, 3/2, 5/4, 7/4 ]; [ 3, 4, 5 ]; 256 x ).
a(n) ~ 3*2^(8*n+3/2)/(Pi^(3/2)*n^(15/2)). - Vaclav Kotesovec, Nov 18 2016
E.g.f.: 3F3(1/4,1/2,3/4; 2,3,4; 256*x) - 1. - Ilya Gutkovskiy, Oct 13 2017
(n+3)*(n+2)*(n+1)*a(n) -8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1)=0. - R. J. Mathar, Mar 04 2018

a(0)=1 prepended by Seiichi Manyama, Nov 23 2018

A005791 5-dimensional Catalan numbers.

Original entry on oeis.org

1, 1, 42, 6006, 1662804, 701149020, 396499770810, 278607172289160, 231471904322784840, 219738059326729823880, 232553551737813227594400, 269396678720275351794712800, 336839101096824285057473785200, 449620757769949216266129125515200

Offset: 0

N. J. A. Sloane

Number of standard tableaux of shape (n,n,n,n,n). - Emeric Deutsch, May 13 2004

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Snover, Stephen L.; Troyer, Stephanie F.; A four-dimensional Catalan formula. Proceedings of the Nineteenth Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, MB, 1989). Congr. Numer. 75 (1990), 123-126.

Alois P. Heinz, Table of n, a(n) for n = 0..294
Shalosh B. Ekhad and Doron Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux. Also arXiv preprint arXiv:1202.6229, 2012. - _N. J. A. Sloane_, Jul 07 2012
K. Gorska and K. A. Penson, Multidimensional Catalan and related numbers as Hausdorff moments, arXiv preprint arXiv:1304.6008 [math.CO], 2013, and Prob. Math. Stat. 33 (2) (2013) 265-274.
S. Snover, Letter to N. J. A. Sloane, May 1991
S. F. Troyer & S. L. Snover, m-Dimensional Catalan numbers, Preprint, 1989. (Annotated scanned copy)

A row of A060854.

a:= n-> (5*n)! * mul(i!/(n+i)!, i=0..4):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 23 2017

Table[288*(5*n)!/(n!*(n+1)!*(n+2)!*(n+3)!*(n+4)!), {n, 1, 20}] (* Vaclav Kotesovec, Nov 18 2016 *)

a(n) = 0!*1!*..*(k-1)! *(k*n)! / ( n!*(n+1)!*..*(n+k-1)! ) for k=5.
(n+4)*(n+3)*(n+2)*(n+1)*a(n) -5*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-1)=0. - R. J. Mathar, Aug 10 2015
G.f.: x*5F4(1,6/5,7/5,8/5,9/5;3,4,5,6;3125*x). - R. J. Mathar, Aug 10 2015
a(n) ~ 72*5^(5*n+1/2)/(Pi^2*n^12). - Vaclav Kotesovec, Nov 18 2016
E.g.f.: 4F4(1/5,2/5,3/5,4/5; 2,3,4,5; 3125*x). - Ilya Gutkovskiy, Oct 13 2017

a(0)=1 prepended by Alois P. Heinz, Jul 23 2017

A007724 Even minus odd extensions of truncated 3 X 2n grid diagram.

Original entry on oeis.org

2, 12, 110, 1274, 17136, 255816, 4124406, 70549050, 1264752060, 23555382240, 452806924752, 8939481277552, 180551099694400, 3719061442253520, 77933728043586630, 1658001861319441050, 35749633305661575300, 780123576993991461000, 17208112644166765652100

Offset: 2

Frank Ruskey

Number of standard tableaux of shapes (n-1,n-1,k), k=0,1,...,n-1. Example: a(3)=12 because there are 2, 5 and 5 standard tableaux of shapes (2,2), (2,2,1) and (2,2,2), respectively. - Emeric Deutsch, May 25 2004
From Joel B. Lewis, Oct 05 2009: (Start)
Also the number of standard shifted Young tableaux of shape (n+1, n, n-1).
Also the number of 2143-avoiding up-down permutations of length 2n - 1. (End)

J. B. Lewis, Pattern Avoidance for Alternating Permutations and Reading Words of Tableaux, Ph. D. Dissertation, Department of Mathematics, MIT, 2012.
F. Ruskey, Generating linear extensions of posets by transpositions, J. Combin. Theory, B 54 (1992), 77-101.
Dennis White, Sign-balanced posets

Cf. A003121.
2143-avoiding up-down permutations of length 2n are given by A005789. - Joel B. Lewis, Oct 05 2009
After corrections, is very similar to A217800.

A007724 := proc(n)
    combinat[multinomial](3*n,n-1,n,n+1)/n/(2*n-1)/(2*n+1) ;
end proc:
seq(A007724(n),n=2..40) ; # R. J. Mathar, Jul 07 2023

Table[(3*n)!/((n-1)!*n!*(n+1)!)/(n*(2*n-1)*(2*n+1)),{n,2,10}] (* Vaclav Kotesovec, Nov 13 2014 *)
Table[(-1)^n HypergeometricPFQ[{-2 - 2 n, -2 n, -2 n - 1}, {2, 3}, 1], {n, 19}] (* Michael De Vlieger, Aug 22 2016 *)

{a(n) = if(n<2, 0, (3*n)!/((2*n+1) * (2*n-1) * (n+1)! * n!^2))}; /* Michael Somos, Jul 04 2020 */

a(n) = multinomial(3n; n-1, n, n+1)/(n(2n-1)(2n+1)).
a(n) ~ 3^(3*n+1/2) / (8*Pi*n^4). - Vaclav Kotesovec, Nov 13 2014
D-finite with recurrence n*(n+1)*(2*n+1)*a(n) -3*(3*n-1)*(2*n-3)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Jul 07 2023

More terms from James Sellers, Dec 24 1999

a(16)-a(18) corrected and a(19)-a(20) added by Alois P. Heinz, Aug 22 2016

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 *)

A217800 Number of alternating permutations on 2n+1 letters that avoid a certain pattern of length 4 (see Lewis, 2012, Appendix, for precise definition).

Original entry on oeis.org

1, 2, 12, 110, 1274, 17136, 255816, 4124406, 70549050, 1264752060, 23555382240, 452806924752, 8939481277552, 180551099694400, 3719061442253520, 77933728043586630, 1658001861319441050, 35749633305661575300, 780123576993991461000, 17208112644166765652100

Offset: 0

N. J. A. Sloane, Oct 12 2012

nonn

1 together with A007724. - Omar E. Pol, Aug 22 2016

K. Gorska and K. A. Penson, Multidimensional Catalan and related numbers as Hausdorff moments, arXiv preprint arXiv:1304.6008 [math.CO], 2013.
J. B. Lewis, Pattern Avoidance for Alternating Permutations and Reading Words of Tableaux, Ph. D. Dissertation, Department of Mathematics, MIT, 2012.

Cf. A007724, A181197, A217799-A217830, A005789, A006480, A007724.

[Factorial(3*n+3)/((4*(n+1)^2-1)*Factorial((n+1))^2*Factorial(n+ 2)): n in [0..20]]; // Vincenzo Librandi, Aug 30 2014

a := n -> (-1)^n*hypergeom([-2-2*n, -2*n, -2*n-1], [2, 3], 1):
seq(round(evalf(a(n), 32)), n=0..20); # Peter Luschny, Aug 29 2014

Table[(3 n + 3)!/((4 (n + 1)^2 - 1) ((n + 1)!)^2 (n + 2)!), {n, 0, 20}] (* Vincenzo Librandi, Aug 30 2014 *)
Table[(-1)^n HypergeometricPFQ[{-2 - 2 n, -2 n, -2 n - 1}, {2, 3}, 1], {n, 0, 20}] (* Michael De Vlieger, Aug 22 2016 *)

a(n) = (3*n+3)!/((4*(n+1)^2-1)*((n+1)!)^2*(n+2)!); \\ Michel Marcus, Aug 10 2014

From Karol A. Penson, Aug 10 2014: (Start)
O.g.f.(in Maple notation): hypergeom([1/2, 1, 4/3, 5/3], [2, 5/2, 3], 27*z);a(n) ~ (1/93312)*sqrt(3)*27^n*(314928*n^4-1644624*n^3+5545260*n^2 -15387660*n+38310503)/(Pi*n^8), for n -> infinity.
Representation of a(n) as the n-th power moment of a positive function on the segment [0,27]:
a(n) = int(x^n*W(x),x=0..27),n=0,1,2..., where
W(x) = 1/(Pi*sqrt(x))+sqrt(x)/Pi-(9/20)*sqrt(3)*2^(1/3)* hypergeom([-2/3, -1/6, 1/3], [2/3, 11/6], (1/27)*x)*x^(1/3)/ (sqrt(Pi)*Gamma(5/6)*Gamma(2/3))-(27/56)*2^(2/3)*Gamma(5/6) *Gamma(2/3)*hypergeom([-1/3, 1/6, 2/3], [4/3, 13/6], (1/27)*x)* x^(2/3)/Pi^(5/2).
W(x) for x->0 has the singularity 1/sqrt(x), W(27)=0.
This is the solution of the Hausdorff moment problem and is unique.
a(n) = (1/2)*(n+3)!/((4*(n+1)^2-1)*(n+1)!)*A005789(n), where A005789(n) are the three-dimensional Catalan numbers (see the Gorska and Penson link).(End)
a(n) = A006480(n+1)/((2+n)*(1+2*n)*(3+2*n)). - Peter Luschny, Aug 15 2014
a(n) = (-1)^n*hypergeom([-2-2*n,-2*n,-2*n-1],[2,3],1). - Peter Luschny, Aug 29 2014
(2*n+3)*(n+2)*(n+1)*a(n) -3*(3*n+2)*(2*n-1)*(3*n+1)*a(n-1)=0. - R. J. Mathar, Jun 14 2016
a(n) ~ 3^(3*n + 7/2) / (8*Pi*n^4). - Vaclav Kotesovec, Jun 09 2019

More terms from Alois P. Heinz, Aug 22 2016

Merged with A241958 by R. J. Mathar, Jul 07 2023

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

A123555 Number of standard Young tableaux of type (n+1,n,n-1).

Original entry on oeis.org

0, 2, 16, 168, 2112, 30030, 466752, 7759752, 135980416, 2485891980, 47052314400, 916847954880, 18311313000960, 373542610526280, 7761573156274560, 163893933165976200, 3510476121410184960, 76151734612882397700, 1670824967127762045600, 37036620104665392010800, 828632324276985756528000

Offset: 0

Amitai Regev (amitai.regev(AT)weizmann.ac.il), Nov 15 2006

For n > 0, a(n) is the number of up-down permutations of length 2n + 1 with no four-term increasing subsequence. Equivalently, this is the number of up-down permutations of length 2n + 1 with no four-term decreasing subsequence; the number of down-up permutations of length 2n + 1 with no four-term increasing subsequence; and the number of down-up permutations of length 2n + 1 with no four-term decreasing subsequence. (An up-down permutation is one whose descent set is {2, 4, 6, ...}.). - Joel B. Lewis, Oct 05 2009

For definition see James and Kerber, Representation Theory of Symmetric Group, Addison-Wesley, 1981, p. 107.

G. C. Greubel, Table of n, a(n) for n = 0..700
Joerg Arndt, The a(3)=168 Young tableaux of shape [4,3,2].
Joel B. Lewis, Pattern avoidance and RSK-like algorithms for alternating permutations and Young tableaux, arXiv:0909.4966 [math.CO], 2009-2011. [_Joel B. Lewis_, Oct 05 2009]
Joel B. Lewis, Pattern Avoidance for Alternating Permutations and Reading Words of Tableaux, Ph. D. Dissertation, Department of Mathematics, MIT, 2012.
Sherry H. F. Yan, On Wilf equivalence for alternating permutations, Elect. J. Combinat.; 20 (2013), #P58.
Index entries for sequences related to Young tableaux.

Cf. A011553.

For up-down permutations of even length, see A005789. [Joel B. Lewis, Oct 05 2009]

f[n_]:=16 (3 n)!/((n-1)! (n+1)! (n+3)!)
(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) Table[ NumberOfTableaux@ {n + 1, n, n - 1}, {n, 0, 17}] (* Robert G. Wilson v *)

for(n=0,25, print1(16*(3*n)!/((n-1)!*(n+1)!*(n+3)!), ", ")) \\ G. C. Greubel, Oct 15 2017

a(n) = 16*(3*n)!/((n-1)!*(n+1)!*(n+3)!).
(n-1)*(n+3)*(n+1)*a(n) -3*n*(3*n-1)*(3*n-2)*a(n-1)=0, n>1. - R. J. Mathar, Aug 10 2015
G.f.: 2x*3F2(5/3,4/3,2;3,5;27x). - R. J. Mathar, Aug 10 2015

cp's OEIS Frontend

A151334 Duplicate of A005789.

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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.

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A039622 Number of n X n Young tableaux.

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A005790 4-dimensional Catalan numbers.

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A005791 5-dimensional Catalan numbers.

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A007724 Even minus odd extensions of truncated 3 X 2n grid diagram.

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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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