A039622 -id:A039622 - 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

A034841 a(n) = (n^2)! / (n!)^n.

Original entry on oeis.org

1, 1, 6, 1680, 63063000, 623360743125120, 2670177736637149247308800, 7363615666157189603982585462030336000, 18165723931630806756964027928179555634194028454000000, 53130688706387569792052442448845648519471103327391407016237760000000000

Offset: 0

Erich Friedman

nonn

The number of arrangements of 1,2,...,n^2 in an n X n matrix such that each row is increasing. - Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 12 2001
a(n) == 0 (mod n!). In fact (n^2)! == 0 (mod (n!)^n) by elementary combinatorics, a better result is (n^2)! == 0 (mod (n!)^(n+1)). - Amarnath Murthy, Jul 13 2005
a(n) is also the number of lattice paths from {n}^n to {0}^n using steps that decrement one component by 1. a(2) = 6: [(2,2), (1,2), (0,2), (0,1), (0,0)], [(2,2), (1,2), (1,1), (0,1), (0,0)], [(2,2), (1,2), (1,1), (1,0), (0,0)], [(2,2), (2,1), (1,1), (0,1), (0,0)], [(2,2), (2,1), (1,1), (1,0), (0,0)], [(2,2), (2,1), (2,0), (1,0), (0,0)]. - Alois P. Heinz, May 06 2013
Given n^2 distinguishable balls and n distinguishable urns, a(n) = the number of ways to place n balls in the i-th urn for all 1 <= i <= n, where n = n_1 + n_2 + ... + n_n. - Ross La Haye, Dec 28 2013

Alois P. Heinz and Tilman Piesk, Table of n, a(n) for n = 0..26 (first 20 terms from Alois P. Heinz)
Noah Lordi, Maedee Trank-Greene, Akira Kyle, and Joshua Combes, Quantum permutation puzzles with indistinguishable particles, arXiv:2410.22287 [quant-ph], 2024. See p. 8.

Cf. A000142, A039622, A229050, A229050, A340590.
Cf. A000984, A006480, A008977, A008978, A022915.
Main diagonal of A089759, A187783.

[Factorial(n^2) / Factorial(n)^n: n in [0..10]]; // Vincenzo Librandi, Oct 29 2014

a:= n-> (n^2)! / (n!)^n:
seq(a(n), n=0..10);  # Alois P. Heinz, Jul 24 2012

Prepend[Table[nn = n^2;nn! Coefficient[Series[(x^n/n!)^n, {x, 0, nn}], x^nn], {n, 1, 15}], 1] (* Geoffrey Critzer, Mar 08 2015 *)

a(n) = (n^2)! / (n!)^n; \\ Michel Marcus, Oct 28 2014

Using a higher order version of Stirling's formula (the "standard" formula appears in A000142) we have the asymptotic expression: a(n) ~ sqrt(2*Pi) * e^(-1/12) * n^(n^2 - n/2 + 1) / (2*Pi)^(n/2). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 13 2001

a(n) = Product_{k=1..n} binomial(k*n, n). - Vaclav Kotesovec, Mar 10 2019

a(0)=1 prepended by Tilman Piesk, Oct 28 2014

A088020 a(n) = (n^2)!.

Original entry on oeis.org

1, 1, 24, 362880, 20922789888000, 15511210043330985984000000, 371993326789901217467999448150835200000000, 608281864034267560872252163321295376887552831379210240000000000

Offset: 0

Hugo Pfoertner, Sep 18 2003

a(n) is the number of ways in which is possible to fill an n X n square matrix with n^2 distinct elements. - Stefano Spezia, Sep 16 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..21
Index entries for sequences related to tic-tac-toe

Cf. A000142 (n!), A000290 (n^2).
Cf. A039622, A179268. - Reinhard Zumkeller, Jul 06 2010
Cf. A272094, A272095, A272164.

List([0..10],n->Factorial(n^2)); # Muniru A Asiru, Sep 17 2018

[Factorial(n^2): n in [0..10]]; // Vincenzo Librandi, May 31 2011

seq(factorial(n^2),n=0..10); # Muniru A Asiru, Sep 17 2018

Table[(n^2)!,{n,0,9}] (* Vladimir Joseph Stephan Orlovsky, May 19 2011 *)

for(n=0,10,print1((n^2)!,",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 22 2006

A107254 a(n) = SF(2n-1)/SF(n-1)^2 where SF = A000178.

Original entry on oeis.org

1, 1, 12, 8640, 870912000, 22122558259200000, 222531556847250309120000000, 1280394777025250130271722799104000000000, 5746332926632566442385615219551212618645504000000000000

Offset: 0

Henry Bottomley, May 14 2005

nonn

Inverse product of all matrix elements of n X n Hilbert matrix M(i,j) = 1/(i+j-1) (i,j = 1..n). - Alexander Adamchuk, Apr 12 2006

The n X n matrix with A(i,j) = 1/(i+j-1)! (i,j = 1..n) has determinant (-1)^floor(n/2)/a(n). - Mikhail Lavrov, Nov 01 2022

			a(3) = 1!*2!*3!*4!*5!/(1!*2!*1!*2!) = 34560/4 = 8640.
n = 2: HilbertMatrix[n,n]
  1/1 1/2
  1/2 1/3
so a(2) = 1 / (1 * 1/2 * 1/2 * 1/3) = 12.
The n X n Hilbert matrix begins:
  1/1 1/2 1/3 1/4  1/5  1/6  1/7  1/8 ...
  1/2 1/3 1/4 1/5  1/6  1/7  1/8  1/9 ...
  1/3 1/4 1/5 1/6  1/7  1/8  1/9 1/10 ...
  1/4 1/5 1/6 1/7  1/8  1/9 1/10 1/11 ...
  1/5 1/6 1/7 1/8  1/9 1/10 1/11 1/12 ...
  1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 ...

Alois P. Heinz, Table of n, a(n) for n = 0..20
Mathematics Stack Exchange, Determinant of a matrix involving factorials.
Eric Weisstein's World of Mathematics, Hilbert Matrix.

Cf. A000178, A002457, A005249, A098118.

A107254:= func< n | n eq 0 select 1 else (&*[Factorial(n+j)/Factorial(j): j in [0..n-1]]) >;
[A107254(n): n in [0..12]]; // G. C. Greubel, Apr 21 2021

a:= n-> mul((n+i)!/i!, i=0..n-1):
seq(a(n), n=0..10);  # Alois P. Heinz, Jul 23 2012

Table[Product[(i+j-1),{i,1,n},{j,1,n}], {n,1,10}] (* Alexander Adamchuk, Apr 12 2006 *)
Table[n!*BarnesG[2n+1]/(BarnesG[n+2]*BarnesG[n+1]), {n,0,12}] (* G. C. Greubel, Apr 21 2021 *)

a = lambda n: prod(rising_factorial(k,n) for k in (1..n))
print([a(n) for n in (0..10)]) # Peter Luschny, Nov 29 2015

a(n) = n!*(n+1)!*(n+2)!*...*(2n-1)!/(0!*1!*2!*3!*...*(n-1)!) = A000178(2n-1)/A000178(n-1)^2 = A079478(n)/A000984(n) = A079478(n-1)*A009445(n-1) = A107252(n)*A000142(n) = A088020(n)/A039622(n).
a(n) = 1/Product_{j=1..n} ( Product_{i=1..n} 1/(i+j-1) ). - Alexander Adamchuk, Apr 12 2006
a(n) = 2^(n*(n-1)) * A136411(n) for n > 0 . - Robert Coquereaux, Apr 06 2013
a(n) = A136411(n) * A053763(n) for n > 0. [Following remark from Robert Coquereaux] - M. F. Hasler, Apr 06 2013
a(n) ~ A * 2^(2*n^2-1/12) * n^(n^2+1/12) / exp(3*n^2/2+1/12), where A = 1.28242712910062263687534256886979... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Feb 10 2015
a(n) = Product_{k=1..n} rf(k,n) where rf denotes the rising factorial. - Peter Luschny, Nov 29 2015
a(n) = (n! * G(2*n+1))/(G(n+1)*G(n+2)), where G(n) is the Barnes G - function. - G. C. Greubel, Apr 21 2021

A323529 Number of strict square plane partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 5, 7, 11, 13, 19, 23, 31, 37, 47, 55, 69, 79, 95, 109, 129, 145, 169, 189, 217, 241, 273, 301, 339, 371, 413, 451, 499, 541, 595, 643, 703, 757, 823, 925, 999, 1107, 1229, 1387, 1559, 1807, 2071, 2453, 2893, 3451, 4109, 5011

Offset: 0

Gus Wiseman, Jan 17 2019

nonn

			The a(12) = 5 strict square plane partitions:
  [12]
.
  [1 2] [1 2] [1 3] [1 4]
  [3 6] [4 5] [2 6] [2 5]
The a(15) = 13 strict square plane partitions:
  [15]
.
  [7 5] [8 4] [9 3] [6 5] [7 4] [9 2] [6 4] [7 3] [8 2] [6 3] [6 3] [7 2]
  [2 1] [2 1] [2 1] [3 1] [3 1] [3 1] [3 2] [4 1] [4 1] [4 2] [5 1] [5 1]

Alois P. Heinz, Table of n, a(n) for n = 0..7500

Cf. A000219, A008289, A039622, A047966, A089299, A101509, A319066, A323429, A323434, A323522, A323530.

h:= proc(n) h(n):= (n^2)!*mul(k!/(n+k)!, k=0..n-1) end:
b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, `if`(issqr(t), h(isqrt(t)), 0),
         b(n, i-1, t) +b(n-i, min(n-i, i-1), t+1)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 24 2019

Table[Sum[Length[Select[Union[Sort/@Tuples[Reverse/@IntegerPartitions[#,{Length[ptn]}]&/@ptn]],UnsameQ@@Join@@#&&And@@OrderedQ/@Transpose[#]&]],{ptn,IntegerPartitions[n]}],{n,30}]
(* Second program: *)
h[n_] := (n^2)! Product[k!/(k+n)!, {k, 0, n-1}];
b[n_, i_, t_] := b[n, i, t] = If[n > i(i+1)/2, 0, If[n == 0, If[IntegerQ[ Sqrt[t]], h[Sqrt[t]], 0], b[n-i, Min[n-i, i-1], t+1] + b[n, i-1, t]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 70] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

a(n) = Sum_{j>=0} A039622(j) * A008289(n,j^2). - Alois P. Heinz, Jan 24 2019

More terms from Alois P. Heinz, Jan 24 2019

A374514 Number of n X n matrices whose values cover an initial interval of positive integers and whose rows and columns have values which are strictly increasing.

Original entry on oeis.org

1, 1, 3, 197, 732963, 289599115433, 19454710000290140631, 324252739440855086589750626125, 1839663535877691613435674541258128354870051, 4664717625821787781559533555514908690826684467996898799881, 6714190347498763079980307954946450922919624466513063316268554904936722083543

Offset: 0

Andrew Howroyd, Sep 16 2024

nonn

			The a(2) = 3 matrices are:
  [1 2]    [1 2]    [1 3]
  [2 3]    [3 4]    [2 4]

Ludovic Schwob, Table of n, a(n) for n = 0..20
Andrew Howroyd, PARI Program, Sep 2024.

Main diagonal of A374985.

Cf. A039622 (case all values also distinct), A068942, A376162 (case for symmetric matrices).

\\ See Links section for program file.
vector(8, n, A374514(n-1))

A060856 Multi-dimensional Catalan numbers: diagonal T(n,n+2) of A060854.

Original entry on oeis.org

1, 14, 6006, 140229804, 278607172289160, 67867669180627125604080, 2760171874087743799855959353857200, 24486819823897171791550434989846505231774984000, 59986874261544072491135645330451363110127974096720977464312000

Offset: 1

R. H. Hardin, May 03 2001

G. C. Greubel, Table of n, a(n) for n = 1..29

Cf. A060854, A074962.

Cf. A039622.

Table[Product[j!/(n+j)!,{j,0,n+1}]*(n*(n+2))!,{n,1,10}] (* Vaclav Kotesovec, Mar 09 2015 *)

a(n) = 0!*1!*...*(k-1)! *(k*n)! / ( n!*(n+1)!*...*(n+k-1)! ) for k=n+2.
a(n) ~ sqrt(Pi) * exp(n^2/2 + 2*n + 25/12) * n^(n^2 + 2*n + 11/12) / (A * 2^(2*n^2 + 4*n + 17/12)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 09 2015
a(n) = A039622(n+1) / (n+1). - Tom Copeland, May 30 2022

A039917 Number of orderings of 1,2,...,n^2 in an n X n matrix such that each row, each column and both diagonals are increasing.

Original entry on oeis.org

1, 1, 9, 2017, 21569213, 17835527619513, 1677123511579177202174, 24742950249259362969953039657613, 75512002909758683196631913316950684079768626, 60752021865167494642984305761115275381534124800396484901989, 15991585283632910454908797943467512732011897255095362833558749286619895509557

Offset: 1

Floor van Lamoen

nonn

			From _Alois P. Heinz_, Jul 23 2012: (Start)
a(2) = 1:  [1, 3]
           [2, 4].
a(3) = 9:
[1, 4, 7]  [1, 3, 7]  [1, 2, 7]  [1, 4, 6]  [1, 3, 6]
[2, 5, 8]  [2, 5, 8]  [3, 5, 8]  [2, 5, 8]  [2, 5, 8]
[3, 6, 9]  [4, 6, 9]  [4, 6, 9]  [3, 7, 9]  [4, 7, 9]
.
[1, 2, 6]  [1, 4, 6]  [1, 3, 6]  [1, 2, 6]
[3, 5, 8]  [2, 5, 7]  [2, 5, 7]  [3, 5, 7]
[4, 7, 9]  [3, 8, 9]  [4, 8, 9]  [4, 8, 9]. (End)

Alois P. Heinz, Table of n, a(n) for n = 1..14
Index entries for sequences related to Young tableaux.

Cf. A039622, A181191.

b:= proc(l) option remember; local n; n:= nops(l); `if`({l[]}={0},
      1, add(`if`((l[i]-1<>n-i or i=1 or l[i-1]-1<=n-i) and l[i]>
      `if`(i=n, 0, l[i+1]), b(subsop(i=l[i]-1, l)), 0), i=1..n))
    end:
a:= n-> b([n$n]):
seq(a(n), n=1..8);  # Alois P. Heinz, Jul 23 2012

b[l_List] := b[l] = Module[{n = Length[l]}, If[Union[l] == {0}, 1, Sum[If[ (l[[i]]-1 != n-i || i == 1 || l[[i-1]]-1 <= n-i) && l[[i]] > If[i == n, 0, l[[i+1]]], b[ReplacePart[l, i -> l[[i]]-1]], 0], {i, 1, n}]]];
a[n_] := b[Table[n, {n}]];
Array[a, 8] (* Jean-François Alcover, Nov 08 2020, after Alois P. Heinz *)

One more term from Jud McCranie, Aug 09 2001

a(6)-a(13) from Alois P. Heinz, Jul 23 2012

A067700 a(n) = 2(n^2)!Product_{k=0..n-1} k!/(n+k)!.

Original entry on oeis.org

2, 2, 4, 84, 48048, 1402298040, 3343286067469920, 950147368528779758457120, 44162749985403900797695349661715200, 440762756830149092247907829817237094171949712000

Offset: 0

Lekraj Beedassy, Feb 05 2002

G. C. Greubel, Table of n, a(n) for n = 0..30

Cf. A000178, A039622, A079402.

[n eq 0 select 2 else 2*Round(Factorial(n^2)*(&*[ Gamma(j+1)/Gamma(n+j+1): j in [0..n-1]])): n in [0..12]]; // G. C. Greubel, May 04 2021

Table[2*(n^2)!*BarnesG[n+1]^2/BarnesG[2n+1], {n, 0, 12}] (* G. C. Greubel, May 04 2021 *)

[2*factorial(n^2)*product( gamma(j+1)/gamma(n+j+1) for j in (0..n-1) ) for n in (0..12)] # G. C. Greubel, May 04 2021

(a(n)/2)^2 = A079402(n).
a(n) = 2*A039622(n). - Vaclav Kotesovec, Dec 17 2016
a(n) = 2*(n^2)!*BarnesG(n+1)^2/BarnesG(2*n+1), where BarnesG(n) = A000178(n). - G. C. Greubel, May 04 2021

The original definition was unclear (at least to me) and the explicit formula provided did not match the sequence. The new definition was provided by Robert G. Wilson v and is a close match to the beginning of the old version. - N. J. A. Sloane, Feb 10 2002

Edited by Dean Hickerson, Jan 06 2003

A079402 Number of permutations of n^2 distinct integers free of any monotonic increasing or decreasing (n+1)-subsequence.

Original entry on oeis.org

1, 1, 4, 1764, 577152576, 491609948246960400, 2794390432234620616607526201600, 225695005480541203944756162668572542540719673600, 487587121568323060029971679617336086880787102621519060769151477760000

Offset: 0

Dean Hickerson, Jan 06 2003

Conjecture: this is equal to the number of permutations of n^2 distinct integers free of any monotonic increasing or decreasing (n+1)-subsequence. (By the Erdos-Szekeres Theorem, every permutation of n^2+1 distinct integers has such a subsequence.) - Joseph Myers, Jan 04 2003
Claude Lenormand (claude.lenormand(AT)free.fr) confirms this conjecture. - Jan 06, 2002.
a(n) is equal to the number of permutations of n^2 distinct integers having no monotonic sequences of length more than n. - Michael Lugo (mlugo(AT)math.upenn.edu), Mar 25 2009
By Robinson-Schensted correspondance, equals the square of the number of square standard Young tableaux. - Wouter Meeussen, Jan 25 2014

			The case n=2: only a(2)=4 of the 24 permutations of {1,2,3,4} are devoid of any 3-term increasing or decreasing subsequence, namely {2,1,4,3}, {2,4,1,3}, {3,1,4,2}, {3,4,1,2}.

Martin Gardner, Riddles of The Sphinx, MAA, NML vol. 32, 1987, p. 6.
D. E. Knuth, The Art of Computer Programming, Vol. 3: Sorting ang Searching, Addison-Wesley, 1973, p. 69. [From Michael Lugo (mlugo(AT)math.upenn.edu), Mar 25 2009]

Alois P. Heinz, Table of n, a(n) for n = 0..20
Stanley Rabinowitz (proposer), Richard Stanley (solver), Advanced Problem 5641, Amer. Math. Monthly 76 (1969), no. 10, 1153.
Vincent Vatter, An Erdős--Hajnal analogue for permutation classes, arXiv:1511.01076 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Erdos-Szekeres Theorem
Wikipedia, Robinson-Schensted correspondence
Index entries for sequences related to Young tableaux.

Cf. A000178, A039622, A067700.

[n eq 0 select 1 else ( Round(Factorial(n^2)*(&*[ Gamma(j+1)/Gamma(n+j+1): j in [0..n-1]])) )^2: n in [0..10]]; // G. C. Greubel, May 04 2021

a:= n-> ((n^2)! *mul(k!/(n+k)!, k=0..n-1))^2:
seq(a(n), n=0..8);  # Alois P. Heinz, Jan 25 2014

Table[( (n^2)! Product[k!/(n+k)!, {k,0,n-1}] )^2, {n,0,5}] (* Wouter Meeussen, Jan 25 2014 *)
Table[((n^2)!*BarnesG[n+1]^2/BarnesG[2n+1])^2, {n, 0, 10}] (* G. C. Greubel, May 04 2021 *)

[( factorial(n^2)*product( gamma(j+1)/gamma(n+j+1) for j in (0..n-1) ) )^2 for n in (0..10)] # G. C. Greubel, May 04 2021

a(n) = ((n^2)! * Product_{k=0..n-1} k!/(n+k)!)^2.
a(n) = (A067700(n)/2)^2 = A039622(n)^2.
a(n) ~ Pi * n^(2*n^2+11/6) * exp(n^2 + 1/6) / (A^2 * 2^(4*n^2-7/6)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Dec 17 2016
a(n) = ( (n^2)!*BarnesG(n+1)^2/BarnesG(2*n+1) )^2, where BarnesG(n) = A000178(n). - G. C. Greubel, May 04 2021

Better name from Wouter Meeussen, Jan 25 2014

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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A034841 a(n) = (n^2)! / (n!)^n.

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A088020 a(n) = (n^2)!.

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A107254 a(n) = SF(2n-1)/SF(n-1)^2 where SF = A000178.

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A323529 Number of strict square plane partitions of n.

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A374514 Number of n X n matrices whose values cover an initial interval of positive integers and whose rows and columns have values which are strictly increasing.

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A060856 Multi-dimensional Catalan numbers: diagonal T(n,n+2) of A060854.

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A067700 a(n) = 2(n^2)!Product_{k=0..n-1} k!/(n+k)!.