A256553 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).

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

Offset: 0

Author

Alois P. Heinz, Apr 01 2015

Keywords

• nonn
• look
• tabf

Examples

			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;
		

Links

• Alois P. Heinz, Rows n = 0..60, flattened

Crossrefs

Row sums give A060179.

Row lengths give A009490.

Last elements of rows give A000793.

Main diagonal gives A000027.

Cf. A181844, A256067, A256554.

Programs

Maple

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);

Mathematica

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]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jul 15 2021, after Alois P. Heinz *)

Formula

Sum_{k>=0} T(n,k)*A256554(n,k) = A181844(n).

T(n,k) = k for n>0 and 1<=k<=n.