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A215120
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Number T(n,k) of solid standard Young tableaux of n cells and height >= k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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2
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1, 1, 1, 3, 3, 1, 9, 9, 5, 1, 33, 33, 23, 7, 1, 135, 135, 109, 43, 9, 1, 633, 633, 557, 261, 69, 11, 1, 3207, 3207, 2975, 1641, 507, 101, 13, 1, 17589, 17589, 16825, 10503, 3787, 869, 139, 15, 1, 102627, 102627, 100007, 69077, 28205, 7487, 1369, 183, 17, 1
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OFFSET
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0,4
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LINKS
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FORMULA
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T(n,n) = 1, T(n,k) = T(n,k+1) + A214753(n,k) for k<n.
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EXAMPLE
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Triangle T(n,k) begins:
: 1;
: 1, 1;
: 3, 3, 1;
: 9, 9, 5, 1;
: 33, 33, 23, 7, 1;
: 135, 135, 109, 43, 9, 1;
: 633, 633, 557, 261, 69, 11, 1;
: 3207, 3207, 2975, 1641, 507, 101, 13, 1;
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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), [])):
H:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
T:= proc(n, k) option remember; `if`(k=n, 1, T(n, k+1)+ H(n, k)) end:
seq(seq(T(n, k), k=0..n), n=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], {}]];
H[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
T[n_, n_] = 1;
T[n_, k_] := T[n, k] = T[n, k + 1] + H[n, k];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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