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A215086
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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.
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13
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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
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OFFSET
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0,9
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LINKS
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FORMULA
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A(n,k) = Sum_{i=0..k} A214753(n,i).
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EXAMPLE
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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, ...
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MAPLE
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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);
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MATHEMATICA
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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 *)
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CROSSREFS
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Columns k=0-10 give: A000007, A000085, A215087, A320180, A320181, A320182, A320183, A320184, A320185, A320186, A320187.
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KEYWORD
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AUTHOR
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STATUS
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approved
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