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A256553
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Triangle T(n,k) in which the n-th row contains the increasing list of distinct orders of degree-n permutations; n>=0, 1<=k<=A009490(n).
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4
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1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 10, 12, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 20, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 20, 21, 30
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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,k) = k for n>0 and 1<=k<=n.
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EXAMPLE
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Triangle T(n,k) begins:
1;
1;
1, 2;
1, 2, 3;
1, 2, 3, 4;
1, 2, 3, 4, 5, 6;
1, 2, 3, 4, 5, 6;
1, 2, 3, 4, 5, 6, 7, 10, 12;
1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 15;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 20;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 20, 21, 30;
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0 or i=1, x,
b(n, i-1)+(p-> add(coeff(p, x, t)*x^ilcm(t, i),
t=1..degree(p)))(add(b(n-i*j, i-1), j=1..n/i)))
end:
T:= n->(p->seq((h->`if`(h=0, [][], i))(coeff(p, x, i))
, i=1..degree(p)))(b(n$2)):
seq(T(n), n=0..12);
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x,
b[n, i - 1] + Function[p, Sum[Coefficient[p, x, t]*x^LCM[t, i],
{t, 1, Exponent[p, x]}]][Sum[b[n - i*j, i - 1], {j, 1, n/i}]]];
T[n_] := Function[p, Table[Function[h, If[h == 0, Nothing, i]][
Coefficient[p, x, i]], {i, 1, Exponent[p, x]}]][b[n, n]];
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CROSSREFS
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Last elements of rows give A000793.
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KEYWORD
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AUTHOR
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STATUS
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approved
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