A060854 -id:A060854 - cp's OEIS frontend

A067228 Number of rectangular standard Young tableaux with n cells.

1, 2, 2, 4, 2, 12, 2, 30, 44, 86, 2, 1190, 2, 860, 12014, 26886, 2, 184758, 2, 3359202, 2771342, 117574, 2, 327618902, 701149022, 1485802, 828630662, 27350160662, 2, 808310933492, 2, 2979826568702, 291724349282, 259289582, 557214344578322, 2031957220875002, 2

Offset: 1

Naohiro Nomoto, Feb 20 2002

Number of ways to arrange the numbers 1, 2, .., n=i*j into an i*j rectangle so that each row and each column is increasing.

a(p) = 2 for prime p. - Alois P. Heinz, Jul 25 2012

Alois P. Heinz, Table of n, a(n) for n = 1..350

Cf. A000085, A060854, A067231.

Column k=0 of A238707.

with(numtheory):
a:= n-> n! * add(mul(k!/(i+k)!, k=0..(n/i)-1), i=divisors(n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 25 2012

a[n_] := n! * Sum[Product[k!/(i+k)!, {k, 0, n/i-1}], {i, Divisors[n]}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jul 02 2015, after Alois P. Heinz *)

a(n) = n! * Sum_{i|n} Product_{k=0..n/i-1} k!/(i+k)!. - Alois P. Heinz, Jul 25 2012

Better name from Joerg Arndt, Feb 24 2014

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

A067231 Number of Young tableaux with n=ij cells and type ij matrices with i>=j.

Original entry on oeis.org

1, 1, 1, 3, 1, 6, 1, 15, 43, 43, 1, 595, 1, 430, 6007, 25455, 1, 92379, 1, 1679601, 1385671, 58787, 1, 163809451, 701149021, 742901, 414315331, 13675080331, 1, 404155466746, 1, 1489913284351, 145862174641, 129644791, 278607172289161, 1851800127304981, 1

Offset: 1

Naohiro Nomoto, Feb 20 2002

a(p) = 1 for prime p. - Alois P. Heinz, Jul 25 2012

Alois P. Heinz, Table of n, a(n) for n = 1..350

Cf. A000085, A060854, A067228.

with(numtheory):
a:= n-> n!*add(mul(k!/(i+k)!, k=0..n/i-1),
        i=select(d-> is(d>=sqrt(n)), divisors(n))):
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 25 2012

a[n_] := n!*Sum[Product[k!/(i+k)!, {k, 0, n/i-1}], {i, Select[Divisors[n], # >= Sqrt[n]&]}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 23 2017, translated from Maple *)

a(n) = number of ways to arrange the numbers 1, 2, .., n=i*j in i*j matrices so that each row and each column is increasing. Here i and j satisfy i >= j.

a(n) = n! * Sum_{i|n, i>=sqrt(n)} Product_{k=0..n/i-1} k!/(i+k)!. - Alois P. Heinz, Jul 25 2012

A211400 Rectangular array, read by upward diagonals: T(n,m) is the number of Young tableaux that can be realized as the ranks of the outer sums a_i + b_j where a = (a_1, ... a_n) and b = (b_1, ... b_m) are real monotone vectors in general position (all sums different).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 14, 36, 14, 1, 1, 42, 295, 295, 42, 1, 1, 132, 2583, 6660, 2583, 132, 1, 1, 429, 23580

Offset: 1

Colin Mallows, Feb 08 2013

Alternatively, that can be realized as the ranks of the outer products a_i b_j where a = (a_1, ... a_n) and b = (b_1, ... b_m) are real positive monotone vectors.
The entries at T(2,n) and T(m,2) are Catalan numbers (A000108).
The original version of this sequence was
1  1  1  1  1   1   1 ...
1  2  5 14 42 132 428 ...
1  5 24 77 ...
1 14 77 ...
1 42 ...
...
but some of the later entries seem to be incorrect. - Robert J. Vanderbei, Jan 09 2015

			The vectors a = (0,2) and b = (0,4,5) give the outer sums
0  4  5  which have ranks  1  3  4
2  6  7                    2  5  6
which is one of the five 2 X 3 Young tableaux.
One of the 18 3 X 3 tableaux that cannot be realized as a set of outer sums
is  1  2  6
    3  5  7
    4  8  9.
The array begins
1      1      1      1      1      1      1      1      1 ...
1      2      5     14     42    132    429   1430   4862 ... (A000108)
1      5     36    295   2583  23580 221680    ... (A255489)
1     14    295   6660    ...
1     42   2583    ...
1    132  23580    ...
1    429 221680    ...
1   1430   ...
1   4862   ...
...

Federico Castillo and Jean-Philippe Labbé, Lineup polytopes of product of simplices, arXiv:2306.00082 [math.CO], 2023.
C. Mallows, R. J. Vanderbei, Which Young Tableaux Can Represent an Outer Sum?, J. Int. Seq. 18 (2015) #15.9.1.
Robert J. Vanderbei, Solutions for the 3 X 3 case
Robert J. Vanderbei, Solutions for the 3 X 4 case
Robert J. Vanderbei, Solutions for the 4 X 4 case

Cf. A060854, A000108, A255489.

Corrected and extended by Robert J. Vanderbei, Jan 09 2015

A215287 Number of permutations of 0..floor((n*3-1)/2) on even squares of an n X 3 array such that each row and column of even squares is increasing.

Original entry on oeis.org

1, 3, 10, 30, 140, 420, 2310, 6930, 42042, 126126, 816816, 2450448, 16628040, 49884120, 350574510, 1051723530, 7595781050, 22787343150, 168212023980, 504636071940, 3792416540640, 11377249621920, 86787993910800, 260363981732400, 2011383287449200

Offset: 1

R. H. Hardin, Aug 07 2012

nonn

Also Schröder paths of length n having floor(n/2) peaks. - Peter Luschny, Sep 30 2018

			Some solutions for n=5:
  0 x 4   0 x 5   1 x 3   0 x 1   0 x 3   1 x 4   0 x 2
  x 3 x   x 1 x   x 0 x   x 4 x   x 2 x   x 0 x   x 1 x
  1 x 5   2 x 6   2 x 5   2 x 3   1 x 6   2 x 5   3 x 5
  x 7 x   x 3 x   x 6 x   x 6 x   x 5 x   x 6 x   x 6 x
  2 x 6   4 x 7   4 x 7   5 x 7   4 x 7   3 x 7   4 x 7

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

Column 3 of A215292.

Cf. A060854, A060693.

[(n-(n div 2)+1)*Factorial(2*n-(n div 2)) / (Factorial(n-(n div 2) +1)^2*Factorial((n div 2))): n in [1..30]]; // Vincenzo Librandi, Oct 01 2018

T := (n, k) -> (n-k+1)*(2*n-k)!/((n-k+1)!^2*k!):
a := n -> T(n, floor(n/2)): seq(a(n), n = 1..23); # Peter Luschny, Sep 30 2018

Table[(n - Floor[n/2] + 1) (2 n - Floor[n/2])! / ((n -Floor[n/2] + 1)!^2 Floor[n/2]!), {n, 1, 30}] (* Vincenzo Librandi, Oct 01 2018 *)

f3 = floor((n+1)/2); f4 = floor(n/2);
a(n) = A060854(2,f3)*A060854(1,f4)*binomial(2*f3+1*f4,2*f3).
a(n) = (n - f + 1)*(2*n - f)! / ((n - f + 1)!^2 * f!) where f = floor(n/2). - Peter Luschny, Sep 30 2018

A215290 Number of permutations of 0..floor((n*6-1)/2) on even squares of an nX6 array such that each row and column of even squares is increasing.

Original entry on oeis.org

1, 20, 420, 23100, 1051050, 85765680, 5703417720, 577185873264, 48236247979920, 5595404765670720, 545152292883918720, 69506917342699636800, 7562187114225380722800, 1033498905610802032116000

Offset: 1

R. H. Hardin Aug 07 2012

nonn

Column 6 of A215292

			Some solutions for n=4
..0..x..2..x..6..x....1..x..6..x..7..x....1..x..2..x..7..x....0..x..2..x..6..x
..x..1..x..3..x..5....x..0..x..2..x..4....x..0..x..4..x.10....x..3..x..5..x..7
..4..x..9..x.10..x....9..x.10..x.11..x....3..x..6..x..8..x....1..x..9..x.11..x
..x..7..x..8..x.11....x..3..x..5..x..8....x..5..x..9..x.11....x..4..x..8..x.10

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

f3=floor((n+1)/2)
f4=floor(n/2)
a(n) = A060854(3,f3)*A060854(3,f4)*binomial(3*f3+3*f4,3*f3)

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

A062528 Number of ways to fill an n X n matrix with numbers 1, 2, ..., n*n such that each row and each column is decreasing or increasing.

Original entry on oeis.org

1, 24, 736, 486200, 15717093608, 41301356272912040, 12775545227876350768099688, 645382645785863222088428068265093848, 6899505383250315051751361998008633233501722290328

Offset: 1

Floor van Lamoen, Jul 10 2001

From Jon E. Schoenfield, Aug 16 2009: (Start)
Number the rows of the matrix as 1..n from bottom to top, and the columns as 1..n from left to right. For i=1..n, let r(i)=1 if the i-th row increases toward the right, -1 if it decreases; let R be the number of runs of consecutive rows having the same r value, and let LR(k) be the length of the k-th run, for k=1..R. Similarly, for i=1..n, let c(i)=1 if the i-th column increases toward the top, -1 if it decreases; let C be the number of runs of consecutive columns having the same c value, and let LC(k) be the length of the k-th run, for k=1..C.
For each of the 4 possible combinations of the vectors r and c in which R=C=1, the number of solutions is A039622(n). In any combination where R=1 and C > 1, the matrix can be partitioned into C rectangular sections (the k-th one being LC(k) columns wide and n rows high); the numbers 1 through LC(1)*n must be placed in the first section, the next LC(2)*n numbers in the second section, etc., so the total number of solutions is Product_{k=1..C} T(LC(k),n), where T(m,n) is as defined at A060854; similarly, if C=1 and R > 1, the number of solutions is Product_{k=1..R} T(LR(k),n).
In any combination where R=2, C=2, and r(1)=c(1), the matrix can be partitioned into 4 rectangular sections, with the lower left and lower right sections covering rows 1..LR(1), the upper left and upper right covering the remaining rows, the lower and upper left covering columns 1..LC(1), and the lower and upper right covering the remaining columns. Then, if r(1)=c(1)=1, the numbers 1 through LR(1)*LC(1) + LR(2)*LC(2) must be apportioned between the lower left and upper right sections; if r(1)=c(1)=-1, they must be apportioned between the other two sections. Either way, the number of solutions for such a combination of the vectors r and c is binomial(LR(1)*LC(1) + LR(2)*LC(2), LR(1)*LC(1)) * binomial(LR(1)*LC(2) + LR(2)*LC(1), LR(1)*LC(2)) * T(LR(1), LC(1)) * T(LR(1), LC(2)) * T(LR(2), LC(1)) * T(LR(2), LC(2)).
No solutions exist where R=C=2 and r(1) != c(1), nor are there any solutions where R=2 and C > 2, R > 2 and C=2, or R > 2 and C > 2. (End)

			a(2) = 4! = 24 because every arrangement of the four elements of a 2 X 2 matrix satisfies the conditions.

Cf. A039622.

a(3) changed from 5320 to 736 by R. H. Hardin, Feb 17 2002

More terms from Jon E. Schoenfield, Aug 15 2009

A177847 Array T(n,m)= (nm)!Beta(n, m) read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 6, 60, 60, 6, 24, 2016, 12096, 2016, 24, 120, 120960, 7983360, 7983360, 120960, 120, 720, 11404800, 12454041600, 149448499200, 12454041600, 11404800, 720, 5040, 1556755200, 38109367296000, 8688935743488000, 8688935743488000

Offset: 1

Roger L. Bagula, May 14 2010

Beta(x,y) = Gamma(x)*Gamma(y)/Gamma(x+y).

			The array starts in row n=1 as:
1,       1,           2,                6,                      24, ...
1,       4,          60,             2016,                  120960, ...
2,      60,       12096,          7983360,             12454041600, ...
6,    2016,     7983360,     149448499200,        8688935743488000, ...
24, 120960, 12454041600, 8688935743488000, 24620968322747596800000, ...

Cf. A060854.

A177847 := proc(n,m) (n*m)!*Beta(n,m) ; end proc:
seq (seq (A177847(n, 1+d-n), n=1..d), d=1..10);

t[n_, m_] = (n*m)!*Beta[n, m];
a = Table[Table[t[n, m], {m, 1, 10}], {n, 1, 10}];
Table[Table[a[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}];
Flatten[%]

T(n,m) = Gamma(n*m+1)*Gamma(n)*Gamma(m)/Gamma(n+m).
T(1,m) = A000142(m-1).
T(n,m) = T(m,n).

cp's OEIS Frontend

A067228 Number of rectangular standard Young tableaux with n cells.

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A321977 8-dimensional Catalan numbers.

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A067231 Number of Young tableaux with n=i*j cells and type i*j matrices with i>=j.

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A211400 Rectangular array, read by upward diagonals: T(n,m) is the number of Young tableaux that can be realized as the ranks of the outer sums a_i + b_j where a = (a_1, ... a_n) and b = (b_1, ... b_m) are real monotone vectors in general position (all sums different).

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A215287 Number of permutations of 0..floor((n*3-1)/2) on even squares of an n X 3 array such that each row and column of even squares is increasing.

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Programs

Magma

Maple

Mathematica

Formula

A215290 Number of permutations of 0..floor((n*6-1)/2) on even squares of an nX6 array such that each row and column of even squares is increasing.

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Comments

Examples

Links

Formula

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

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Programs

Magma

Mathematica

Sage

Formula

A321978 9-dimensional Catalan numbers.

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A067231 Number of Young tableaux with n=ij cells and type ij matrices with i>=j.

A177847 Array T(n,m)= (nm)!Beta(n, m) read by antidiagonals.