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

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

Author

Seiichi Manyama, Nov 17 2018

Keywords

• nonn
• tabl

Examples

			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;
		

Links

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

Crossrefs

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

Programs

Magma

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

Mathematica

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 *)

Sage

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

Formula

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