A215204 Number A(n,k) of solid standard Young tableaux of cylindrical shape lambda X k, where lambda ranges over all partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 4, 4, 5, 1, 1, 10, 26, 10, 7, 1, 1, 28, 276, 258, 26, 11, 1, 1, 84, 3740, 14318, 3346, 76, 15, 1, 1, 264, 58604, 1161678, 1214358, 54108, 232, 22, 1, 1, 858, 1010616, 118316062, 741215012, 150910592, 1054256, 764, 30

Offset: 0

Author

Alois P. Heinz, Aug 05 2012

Keywords

• nonn
• tabl

Examples

			Square array A(n,k) begins:
:  1,  1,     1,         1,            1,                1, ...
:  1,  1,     1,         1,            1,                1, ...
:  2,  2,     4,        10,           28,               84, ...
:  3,  4,    26,       276,         3740,            58604, ...
:  5, 10,   258,     14318,      1161678,        118316062, ...
:  7, 26,  3346,   1214358,    741215012,     620383261034, ...
: 11, 76, 54108, 150910592, 840790914296, 7137345113624878, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..15, flattened
• S. B. Ekhad and D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
• Wikipedia, Young tableau

Crossrefs

Columns k=0-5 give: A000041, A000085, A215266, A290202, A290214, A290274.

Rows n=0+1, 2-5 give: A000012, 2*A000108, 2*A005789 + A006335, 2*A005790 + 2*A213978 + A114714, 2*A005791 + 2*A215220 + 2*A213932 + A214638.

Main diagonal gives A290225.

Programs

Maple

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

Mathematica

b[l_] := b[l] = With[{m = Length[l]}, If[Union[l // Flatten] ~Complement~ {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}]]];
g[n_, i_, k_, l_] := If[n == 0 || i == 1, b[Table[k, {#}]& /@ Join[l, Table[1, {n}]]], Sum[g[n - i*j, i - 1, k, Join[l, Table[i, {j}]]], {j, 0, n/i}]];
A[n_, k_] := g[n, n, k, {}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Sep 24 2022, after Alois P. Heinz *)