A214824 -id:A214824 - cp's OEIS frontend

A214722 Number A(n,k) of solid standard Young tableaux of shape [[{n}^k],[n]]; square array A(n,k), n>=0, k>=1, read by antidiagonals.

1, 1, 1, 1, 2, 2, 1, 3, 16, 5, 1, 4, 91, 192, 14, 1, 5, 456, 5471, 2816, 42, 1, 6, 2145, 143164, 464836, 46592, 132, 1, 7, 9724, 3636776, 75965484, 48767805, 835584, 429, 1, 8, 43043, 91442364, 12753712037, 55824699632, 5900575762, 15876096, 1430

Offset: 0

Alois P. Heinz, Jul 26 2012

			Square array A(n,k) begins:
   1,     1,        1,           1,              1,                 1, ...
   1,     2,        3,           4,              5,                 6, ...
   2,    16,       91,         456,           2145,              9724, ...
   5,   192,     5471,      143164,        3636776,          91442364, ...
  14,  2816,   464836,    75965484,    12753712037,     2214110119572, ...
  42, 46592, 48767805, 55824699632, 70692556053053, 98002078234748974, ...

Alois P. Heinz, Antidiagonals n = 0..20, flattened
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Columns k=1-4 give: A000108, A006335, A213978, A215220.
Rows n=0-3 give: A000012, A000027, A214824, A211505.
A(n,n) gives A258583.
Cf. A213932, A214637, A214631, A258586.

b:= proc(l) option remember; local m; m:= nops(l);
      `if`({map(x-> x[], l)[]}={0}, 1, add(add(`if`(l[i][j]>
      `if`(i=m or nops(l[i+1])
      `if`(nops(l[i])=j, 0, l[i][j+1]), b(subsop(i=subsop(
       j=l[i][j]-1, l[i]), l)), 0), j=1..nops(l[i])), i=1..m))
    end:
A:= (n, k)-> b([[n$k], [n]]):
seq(seq(A(n, 1+d-n), n=0..d), d=0..10);

b[l_List] := b[l] = With[{m = Length[l]}, If[Union[Flatten[l]] == {0}, 1, Sum[Sum[If[l[[i, j]] > If[i == m || Length[l[[i+1]]] < j, 0, l[[i+1, j]]] && l[[i, j]] > If[Length[l[[i]]] == j, 0, l[[i, j+1]]], b[ReplacePart[l, i -> ReplacePart[l[[i]], j -> l[[i, j]] - 1]]], 0], {j, 1, Length[l[[i]]]}], {i, 1, m}]] ]; a[n_, k_] := b[{Array[n&, k], {n}}]; Table[Table[a[n, 1+d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 17 2013, translated from Maple *)

A259101 Square array read by antidiagonals arising in the enumeration of corners.

Original entry on oeis.org

1, 2, 2, 5, 16, 5, 14, 91, 91, 14, 42, 456, 936, 456, 42, 132, 2145, 7425, 7425, 2145, 132, 429, 9724, 50765, 85800, 50765, 9724, 429, 1430, 43043, 315315, 805805, 805805, 315315, 43043, 1430, 4862, 187408, 1831648, 6584032, 9962680, 6584032, 1831648, 187408, 4862, 16796, 806208, 10127988, 48674808, 103698504, 103698504, 48674808, 10127988, 806208, 16796

Offset: 0

N. J. A. Sloane, Jun 22 2015

See Kreweras (1979) for precise definition.

			The first few antidiagonals are:
    1,
    2,    2,
    5,   16,    5,
   14,   91,   91,   14,
   42,  456,  936,  456,   42,
  132, 2145, 7425, 7425, 2145, 132,
  ...

G. Kreweras, Sur les extensions linéaires d'une famille particulière d'ordres partiels, Discrete Math., 27 (1979), 279-295.
G. Kreweras, Sur les extensions linéaires d'une famille particulière d'ordres partiels, Discrete Math., 27 (1979), 279-295. (Annotated scanned copy)

The first row and column of the array are the Catalan numbers A000108.

The second row and column are A214824.

a[x_, y_] := (2(2x+2y+1)!(x^2+3x*y+y^2+4x+4y+3)) / (x!(x+1)!y!(y+1)!(x+y+1)(x+y+2)(x+y+3));
Table[Table[a[x-y, y], {y, 0, x}] // Reverse, {x, 0, 9}] // Flatten (* Jean-François Alcover, Aug 11 2017 *)

Kreweras gives an explicit formula for the general term (see bottom display on page 291).

More terms from Jean-François Alcover, Aug 11 2017

cp's OEIS Frontend

A214722 Number A(n,k) of solid standard Young tableaux of shape [[{n}^k],[n]]; square array A(n,k), n>=0, k>=1, read by antidiagonals.

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