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A215086
Number A(n,k) of solid standard Young tableaux of n cells and height <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
13
1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 4, 0, 1, 1, 3, 8, 10, 0, 1, 1, 3, 9, 26, 26, 0, 1, 1, 3, 9, 32, 92, 76, 0, 1, 1, 3, 9, 33, 126, 372, 232, 0, 1, 1, 3, 9, 33, 134, 564, 1566, 764, 0, 1, 1, 3, 9, 33, 135, 622, 2700, 7086, 2620, 0, 1, 1, 3, 9, 33, 135, 632, 3106, 13802, 33550, 9496, 0
OFFSET
0,9
LINKS
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229 [math.CO], 2012
Wikipedia, Young tableau
FORMULA
A(n,k) = Sum_{i=0..k} A214753(n,i).
EXAMPLE
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, 1, 1, ...
0, 2, 3, 3, 3, 3, 3, 3, ...
0, 4, 8, 9, 9, 9, 9, 9, ...
0, 10, 26, 32, 33, 33, 33, 33, ...
0, 26, 92, 126, 134, 135, 135, 135, ...
0, 76, 372, 564, 622, 632, 633, 633, ...
0, 232, 1566, 2700, 3106, 3194, 3206, 3207, ...
MAPLE
b:= proc(n, k, l) option remember; `if`(n=0, 1,
b(n-1, k, [l[], [1]])+ add(`if`(i=1 or nops(l[i])<nops(l[i-1]),
b(n-1, k, subsop(i=[l[i][], 1], l)), 0)+ add(`if`(l[i][j]<k and
(i=1 or l[i][j]<l[i-1][j]) and (j=1 or l[i][j]<l[i][j-1]),
b(n-1, k, subsop(i=subsop(j=l[i][j]+1, l[i]), l)), 0),
j=1..nops(l[i])), i=1..nops(l)))
end:
A:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), b(n, min(n, k), [])):
seq(seq(A(n, d-n), n=0..d), d=0..10);
MATHEMATICA
b[n_, k_, l_] := b[n, k, l] = If[n==0, 1, b[n-1, k, Append[l, {1}]] + Sum[If[i==1 || Length[l[[i]]]<Length[l[[i-1]]], b[n-1, k, ReplacePart[l, i -> Append[l[[i]], 1]]], 0] + Sum[If[l[[i, j]]<k && (i==1 || l[[i, j]] < l[[i-1, j]]) && (j==1 || l[[i, j]]<l[[i, j-1]]), b[n-1, k, ReplacePart[l, i -> ReplacePart[ l[[i]], j -> l[[i, j]]+1]]], 0], {j, 1, Length[l[[i]]]} ], {i, 1, Length[l]}]]; A[n_, k_] := If[k==0, If[n==0, 1, 0], b[n, Min[n, k], {}]]; Table[A[n, d-n], {d, 0, 11}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 26 2017, after Alois P. Heinz *)
CROSSREFS
Rows n=0-1 give: A000012, A057427.
Main diagonal gives A207542.
Cf. A214753.
Sequence in context: A292521 A350828 A354668 * A261440 A295684 A276890
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Aug 02 2012
STATUS
approved