A060854 -id:A060854 - cp's OEIS frontend

A005790 4-dimensional Catalan numbers.

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

A215292 T(n,k)=Number of permutations of 0..floor((n*k-1)/2) on even squares of an nXk array such that each row and column of even squares is increasing.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 6, 10, 6, 1, 1, 10, 30, 30, 10, 1, 1, 20, 140, 280, 140, 20, 1, 1, 35, 420, 2100, 2100, 420, 35, 1, 1, 70, 2310, 23100, 60060, 23100, 2310, 70, 1, 1, 126, 6930, 210210, 1051050, 1051050, 210210, 6930, 126, 1, 1, 252, 42042, 2522520

Offset: 1

R. H. Hardin Aug 07 2012

Table starts
.1...1.....1........1...........1..............1..................1
.1...2.....3........6..........10.............20.................35
.1...3....10.......30.........140............420...............2310
.1...6....30......280........2100..........23100.............210210
.1..10...140.....2100.......60060........1051050...........42882840
.1..20...420....23100.....1051050.......85765680.........5703417720
.1..35..2310...210210....42882840.....5703417720......2061378118800
.1..70..6930..2522520...814773960...577185873264....337653735859440
.1.126.42042.25729704.41227562376.48236247979920.173457547735792320

			Some solutions for n=5 k=4
..1..x..2..x....0..x..6..x....1..x..6..x....1..x..4..x....0..x..6..x
..x..0..x..4....x..3..x..4....x..0..x..3....x..0..x..3....x..1..x..2
..3..x..8..x....1..x..7..x....4..x..7..x....2..x..7..x....4..x..7..x
..x..6..x..7....x..5..x..8....x..2..x..8....x..5..x..6....x..3..x..9
..5..x..9..x....2..x..9..x....5..x..9..x....8..x..9..x....5..x..8..x

R. H. Hardin, Table of n, a(n) for n = 1..1000

Column 2 is A001405

f1=floor((k+1)/2)
f2=floor(k/2)
f3=floor((n+1)/2)
f4=floor(n/2)
T(n,k)=A060854(f1,f3)*A060854(f2,f4)*binomial(f1*f3+f2*f4,f1*f3)

A229345 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component or all components by the same positive integer; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 7, 22, 4, 1, 1, 25, 248, 188, 8, 1, 1, 121, 6506, 11380, 1712, 16, 1, 1, 721, 292442, 2359348, 577124, 16098, 32, 1, 1, 5041, 19450082, 1088626684, 991365512, 30970588, 154352, 64, 1

Offset: 0

Alois P. Heinz, Sep 24 2013

			A(2,2) = 22: [(2,2),(1,1),(0,0)], [(2,2),(1,1),(0,1),(0,0)], [(2,2),(1,1),(1,0),(0,0)], [(2,2),(0,0)], [(2,2),(1,2),(0,1),(0,0)], [(2,2),(1,2),(0,2),(0,1),(0,0)], [(2,2),(1,2),(0,2),(0,0)], [(2,2),(1,2),(1,1),(0,0)], [(2,2),(1,2),(1,1),(0,1),(0,0)], [(2,2),(1,2),(1,1),(1,0),(0,0)], [(2,2),(1,2),(1,0),(0,0)], [(2,2),(0,2),(0,1),(0,0)], [(2,2),(0,2),(0,0)], [(2,2),(2,1),(1,0),(0,0)], [(2,2),(2,1),(1,1),(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),(0,1),(0,0)], [(2,2),(2,1),(2,0),(1,0),(0,0)], [(2,2),(2,1),(2,0),(0,0)], [(2,2),(2,0),(1,0),(0,0)], [(2,2),(2,0),(0,0)].
Square array A(n,k) begins:
  1,  1,     1,        1,            1,                 1, ...
  1,  1,     3,        7,           25,               121, ...
  1,  2,    22,      248,         6506,            292442, ...
  1,  4,   188,    11380,      2359348,        1088626684, ...
  1,  8,  1712,   577124,    991365512,     4943064622568, ...
  1, 16, 16098, 30970588, 453530591824, 25162900228200976, ...

Alois P. Heinz, Antidiagonals n = 0..20, flattened

Columns k=0-3 give: A000012, A011782, A132595(n+1), A229482.
Rows n=0-2 give: A000012, A038507 (for k>1), A229465.
Main diagonal gives: A229346.
Cf. A060854, A227578, A227655, A225094, A210472, A229142, A262809, A263159.

b:= proc(l) option remember; local m; m:= nops(l);
      `if`(m=0 or l[m]=0, 1,
      `if`(m>1, add(b(l-[j$m]), j=1..l[1]), 0)+
      add(add(b(sort(subsop(i=l[i]-j, l))), j=1..l[i]), i=1..m))
    end:
A:= (n, k)-> b([n$k]):
seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Sep 24 2013

b[l_] := b[l] = With[{m = Length[l]}, If[m == 0 || l[[m]] == 0, 1, If[m > 1, Sum[b[l - Array[j&, m]], {j, 1, l[[1]]}],  0] + Sum[Sum[b[Sort[ReplacePart[l, i -> l[[i]] - j]]], {j, 1, l[[i]]}], {i, 1, m}]]]; a[n_, k_] := b[Array[n&, k]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *)

A181204 T(n,k) = 0!1!2!...(k-1)! (nk)! k(k-1)n(n-1) / 2n!(n+1)!...(n+k-1)!

Original entry on oeis.org

0, 0, 0, 0, 4, 0, 0, 30, 30, 0, 0, 168, 756, 168, 0, 0, 840, 16632, 16632, 840, 0, 0, 3960, 360360, 1729728, 360360, 3960, 0, 0, 18018, 7876440, 199536480, 199536480, 7876440, 18018, 0, 0, 80080, 174594420, 25241364720, 140229804000, 25241364720

Offset: 1

R. H. Hardin Oct 10 2010

(Empricial) T(n,k)=Number of nXk matrices containing a defective permutation of 1..n*k in strictly increasing order rowwise and columnwise, with one permutation value omitted and one repeated (see example)
Formula is n*(n-1)*k*(k-1)/2 times n-th k-dimensional Catalan number
Table starts
.0.......0.............0....................0...........................0
.0.......4............30..................168.........................840
.0......30...........756................16632......................360360
.0.....168.........16632..............1729728...................199536480
.0.....840........360360............199536480................140229804000
.0....3960.......7876440..........25241364720.............118949931243000
.0...18018.....174594420........3445446284280..........117015012361447200
.0...80080....3926434512......500598983364480.......129624266420759510400
.0..350064...89492111280....76591644454765440....158211402715245473193600
.0.1511640.2064420294300.12237255920840932800.209298196564031904834960000

			Some solutions for 4X2
..2..4....1..4....1..3....1..3....2..5....1..3....1..3....2..4....1..2....1..3
..3..5....2..5....3..6....2..4....3..6....2..4....2..4....3..6....2..4....2..4
..5..7....6..7....4..7....4..6....4..7....4..5....4..7....5..7....3..5....4..7
..6..8....7..8....5..8....5..8....5..8....7..8....5..8....6..8....6..8....6..8

R. H. Hardin, Table of n, a(n) for n=1..1000

Column 2 is twice A002740(n+1)

Cf. A060854 for permutation without defect.

A215297 T(n,k) = number of permutations of 0..floor((n*k-2)/2) on odd squares of an n X k array such that each row and column of odd squares is increasing.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 6, 6, 6, 1, 1, 10, 30, 30, 10, 1, 1, 20, 70, 280, 70, 20, 1, 1, 35, 420, 2100, 2100, 420, 35, 1, 1, 70, 1050, 23100, 23100, 23100, 1050, 70, 1, 1, 126, 6930, 210210, 1051050, 1051050, 210210, 6930, 126, 1, 1, 252, 18018, 2522520, 14294280

Offset: 1

R. H. Hardin, Aug 07 2012

Table starts
.1...1.....1........1...........1..............1.................1
.1...2.....3........6..........10.............20................35
.1...3.....6.......30..........70............420..............1050
.1...6....30......280........2100..........23100............210210
.1..10....70.....2100.......23100........1051050..........14294280
.1..20...420....23100.....1051050.......85765680........5703417720
.1..35..1050...210210....14294280.....5703417720......577185873264
.1..70..6930..2522520...814773960...577185873264...337653735859440
.1.126.18018.25729704.12547518984.48236247979920.43364386933948080
Even columns match A215292.
The first column is number of symmetric standard Young tableaux of shape (n), the second column is number of symmetric standard Young tableaux of shape (n,n) and the third column is number of symmetric standard Young tableaux of shape (n,n,n). - Ran Pan, May 21 2015

			Some solutions for n=5, k=4:
..x..0..x..4....x..0..x..1....x..1..x..3....x..0..x..6....x..0..x..1
..1..x..2..x....4..x..7..x....0..x..8..x....3..x..5..x....3..x..7..x
..x..3..x..8....x..2..x..3....x..2..x..5....x..1..x..7....x..2..x..5
..6..x..7..x....5..x..9..x....4..x..9..x....4..x..9..x....6..x..8..x
..x..5..x..9....x..6..x..8....x..6..x..7....x..2..x..8....x..4..x..9

R. H. Hardin, Table of n, a(n) for n = 1..1000
Hodge, Jonathan K.; Krines, Mark; Lahr, Jennifer, Preseparable extensions of multidimensional preferences, Order 26, No. 2, 125-147 (2009), Table 1.
Ran Pan, Exercise P, Problem 4, Project P.

Column 2 is A001405. Column 4 is A215288. Column 6 is A215290.

f1=floor(k/2), f2=floor((k+1)/2), f3=floor((n+1)/2), f4=floor(n/2);

T(n,k) = A060854(f1,f3)*A060854(f2,f4)*binomial(f1*f3+f2*f4,f1*f3).

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

A215288 Number of permutations of 0..floor((n*4-1)/2) on even squares of an n X 4 array such that each row and column of even squares is increasing.

Original entry on oeis.org

1, 6, 30, 280, 2100, 23100, 210210, 2522520, 25729704, 325909584, 3585005424, 47117214144, 546896235600, 7383099180600, 89212448432250, 1229149289511000, 15323394475903800, 214527522662653200, 2742051789669912720

Offset: 1

R. H. Hardin, Aug 07 2012

nonn

			Some solutions for n=5
..2..x..4..x....0..x..3..x....2..x..3..x....0..x..5..x....1..x..4..x
..x..0..x..1....x..2..x..5....x..0..x..4....x..1..x..3....x..0..x..6
..3..x..7..x....1..x..6..x....5..x..6..x....6..x..7..x....2..x..5..x
..x..6..x..9....x..4..x..7....x..1..x..8....x..2..x..4....x..3..x..8
..5..x..8..x....8..x..9..x....7..x..9..x....8..x..9..x....7..x..9..x

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 4 of A215292.

f3=floor((n+1)/2),
f4=floor(n/2),
a(n) = A060854(2,f3)*A060854(2,f4)*binomial(2*f3+2*f4,2*f3).

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

A374985 Array read by antidiagonals: T(n,k) is the number of n X k 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 11, 11, 1, 1, 1, 1, 45, 197, 45, 1, 1, 1, 1, 197, 4593, 4593, 197, 1, 1, 1, 1, 903, 126289, 732963, 126289, 903, 1, 1, 1, 1, 4279, 3888343, 155242003, 155242003, 3888343, 4279, 1, 1, 1, 1, 20793, 130016393, 40007492715, 289599115433, 40007492715, 130016393, 20793, 1, 1

Offset: 0

Andrew Howroyd, Sep 16 2024

T(n,k) is the number of normal generalized Young tableaux with all rows and columns strictly increasing whose shape is a rectangle of size n X k (cf. A299968). - Ludovic Schwob, Nov 18 2024

			Array begins:
=====================================================================
n/k | 0 1   2       3           4               5               6 ...
----+----------------------------------------------------------------
  0 | 1 1   1       1           1               1               1 ...
  1 | 1 1   1       1           1               1               1 ...
  2 | 1 1   3      11          45             197             903 ...
  3 | 1 1  11     197        4593          126289         3888343 ...
  4 | 1 1  45    4593      732963       155242003     40007492715 ...
  5 | 1 1 197  126289   155242003    289599115433 723253222084867 ...
  6 | 1 1 903 3888343 40007492715 723253222084867 ...
...
The T(2,3) = 11 matrices are:
  [1 2 3]  [1 2 3]  [1 2 3]  [1 2 3]  [1 2 4]  [1 2 4]
  [2 3 4]  [2 4 5]  [3 4 5]  [4 5 6]  [2 3 5]  [3 4 5]
.
  [1 2 4]  [1 2 5]  [1 3 4]  [1 3 4]  [1 3 5]
  [3 5 6]  [3 4 6]  [2 4 5]  [2 5 6]  [2 4 6]

Andrew Howroyd, Table of n, a(n) for n = 0..230
R. A. Sulanke, Generalizing Narayana and Schroeder Numbers to Higher Dimensions, Electron. J. Combin. 11 (2004), Research Paper 54.

Columns k=1..4 are A000012, A001003, A105124, A374985.
Main diagonal is A374514.
Cf. A060854 (case all values also distinct), A299968.

\\ See PARI link in A374514 for program code.
for(n=0, 7, print(vector(7, k, A374985(n, k-1))))

T(n,k) = T(k,n).

cp's OEIS Frontend

A005790 4-dimensional Catalan numbers.

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A005791 5-dimensional Catalan numbers.

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A215292 T(n,k)=Number of permutations of 0..floor((n*k-1)/2) on even squares of an nXk array such that each row and column of even squares is increasing.

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A229345 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component or all components by the same positive integer; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A181204 T(n,k) = 0!*1!*2!*...*(k-1)! *(n*k)! *k*(k-1)*n*(n-1) / 2*n!*(n+1)!*...*(n+k-1)!

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A215297 T(n,k) = number of permutations of 0..floor((n*k-2)/2) on odd squares of an n X k array such that each row and column of odd squares is increasing.

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A321975 6-dimensional Catalan numbers.

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A181204 T(n,k) = 0!1!2!...(k-1)! (nk)! k(k-1)n(n-1) / 2n!(n+1)!...(n+k-1)!