A214776 Number A(n,k) of standard Young tableaux of shape [n*k,n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 9, 5, 0, 1, 4, 20, 48, 14, 0, 1, 5, 35, 154, 275, 42, 0, 1, 6, 54, 350, 1260, 1638, 132, 0, 1, 7, 77, 663, 3705, 10659, 9996, 429, 0, 1, 8, 104, 1120, 8602, 40480, 92092, 62016, 1430, 0, 1, 9, 135, 1748, 17199, 115101, 451269, 807300, 389367, 4862, 0

Offset: 0

Author

Alois P. Heinz, Jul 28 2012

Keywords

• nonn
• tabl

Comments

A(n,k) is also the number of binary words with n*k 1's and n 0's such that for every prefix the number of 1's is >= the number of 0's. The A(2,2) = 9 words are: 101011, 101101, 101110, 110011, 110101, 110110, 111001, 111010, 111100.

Examples

			Square array A(n,k) begins:
  1,   1,    1,     1,     1,      1,      1, ...
  0,   1,    2,     3,     4,      5,      6, ...
  0,   2,    9,    20,    35,     54,     77, ...
  0,   5,   48,   154,   350,    663,   1120, ...
  0,  14,  275,  1260,  3705,   8602,  17199, ...
  0,  42, 1638, 10659, 40480, 115101, 272272, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140
• Paul Barry, On the Central Antecedents of Integer (and Other) Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.3.
• Wikipedia, Young tableau

Crossrefs

Columns k=0-10 give: A000007, A000108, A174687, A126596, A215541, A215542, A215551, A215552, A215553, A215554, A215555.

Rows n=0-10 give: A000012, A001477, A014107(k+1), A215543, A215544, A215545, A215546, A215547, A215548, A215549, A215550.

Main diagonal gives: A215557.

Programs

Maple

A:= (n, k)-> max(0, binomial((k+1)*n, n)*((k-1)*n+1)/(k*n+1)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

Mathematica

a[n_, k_] := Max[0, Binomial[(k+1)*n, n]*((k-1)*n+1)/(k*n+1)]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Oct 01 2013, after Maple *)

Formula

A(n,k) = max(0, C((k+1)*n,n)*((k-1)*n+1)/(k*n+1)).