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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. 19
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

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.

LINKS

Alois P. Heinz, Antidiagonals n = 0..140

Wikipedia, Young tableau

FORMULA

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

EXAMPLE

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, ...

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-Fran├žois Alcover, Oct 01 2013, after Maple *)

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.

Sequence in context: A237018 A290605 A292913 * A317575 A295653 A146326

Adjacent sequences:  A214773 A214774 A214775 * A214777 A214778 A214779

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jul 28 2012

STATUS

approved

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Last modified May 19 20:41 EDT 2019. Contains 323410 sequences. (Running on oeis4.)