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A210391
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Number A(n,k) of semistandard Young tableaux over all partitions of n with maximal element <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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21
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1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 9, 6, 1, 0, 1, 5, 16, 19, 9, 1, 0, 1, 6, 25, 44, 39, 12, 1, 0, 1, 7, 36, 85, 116, 69, 16, 1, 0, 1, 8, 49, 146, 275, 260, 119, 20, 1, 0, 1, 9, 64, 231, 561, 751, 560, 189, 25, 1, 0
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OFFSET
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0,8
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LINKS
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FORMULA
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G.f. of column k: 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)).
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EXAMPLE
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Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 1, 4, 9, 16, 25, 36, ...
0, 1, 6, 19, 44, 85, 146, ...
0, 1, 9, 39, 116, 275, 561, ...
0, 1, 12, 69, 260, 751, 1812, ...
0, 1, 16, 119, 560, 1955, 5552, ...
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MAPLE
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# First program:
h:= (l, k)-> mul(mul((k+j-i)/(1+l[i] -j +add(`if`(l[t]>=j, 1, 0)
, t=i+1..nops(l))), j=1..l[i]), i=1..nops(l)):
g:= proc(n, i, k, l)
`if`(n=0, h(l, k), `if`(i<1, 0, g(n, i-1, k, l)+
`if`(i>n, 0, g(n-i, i, k, [l[], i]))))
end:
A:= (n, k)-> `if`(n=0, 1, g(n, n, k, [])):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second program:
gf:= k-> 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)):
A:= (n, k)-> coeff(series(gf(k), x, n+1), x, n):
seq(seq(A(n, d-n), n=0..d), d=0..12);
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MATHEMATICA
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(* First program: *)
h[l_, k_] := Product[Product[(k+j-i)/(1+l[[i]]-j + Sum[If[l[[t]] >= j, 1, 0], {t, i+1, Length[l]}]), {j, 1, l[[i]]}], {i, 1, Length[l]}]; g [n_, i_, k_, l_] := If[n == 0, h[l, k], If[i < 1, 0, g[n, i-1, k, l] + If[i > n, 0, g[n-i, i, k, Append[l, i]]]]]; a[n_, k_] := If[n == 0, 1, g[n, n, k, {}]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten
(* second program: *)
gf[k_] := 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)); a[n_, k_] := Coefficient[Series[gf[k], {x, 0, n+1}], x, n]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 09 2013, translated from Maple *)
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CROSSREFS
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Rows n=0-10 give: A000012, A001477, A000290, A005900, A139594, A210427, A210428, A210429, A210430, A210431, A210432.
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KEYWORD
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AUTHOR
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STATUS
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approved
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