A246688 Triangle in which n-th row lists lexicographically ordered increasing lists of parts of all partitions of n into distinct parts.

1, 2, 1, 2, 3, 1, 3, 4, 1, 4, 2, 3, 5, 1, 2, 3, 1, 5, 2, 4, 6, 1, 2, 4, 1, 6, 2, 5, 3, 4, 7, 1, 2, 5, 1, 3, 4, 1, 7, 2, 6, 3, 5, 8, 1, 2, 6, 1, 3, 5, 1, 8, 2, 3, 4, 2, 7, 3, 6, 4, 5, 9, 1, 2, 3, 4, 1, 2, 7, 1, 3, 6, 1, 4, 5, 1, 9, 2, 3, 5, 2, 8, 3, 7, 4, 6, 10

Offset: 1

Author

Alois P. Heinz, Sep 01 2014

Keywords

• nonn
• tabf

Examples

			Triangle begins:
  [1];
  [2];
  [1,2], [3];
  [1,3], [4];
  [1,4], [2,3], [5];
  [1,2,3], [1,5], [2,4], [6];
  [1,2,4], [1,6], [2,5], [3,4], [7];
  [1,2,5], [1,3,4], [1,7], [2,6], [3,5], [8];
  [1,2,6], [1,3,5], [1,8], [2,3,4], [2,7], [3,6], [4,5], [9];
  [1,2,3,4], [1,2,7], [1,3,6], [1,4,5], [1,9], [2,3,5], [2,8], [3,7], [4,6], [10];
		

Links

• Alois P. Heinz, Rows n = 1..32, flattened

Crossrefs

Row lengths are A015723.

Row sums give A066189.

Last elements of rows are A000027.

Cf. A026791, A026793, A118457, A265146.

Programs

Maple

b:= proc(n, i) b(n, i):= `if`(n=0, [[]], `if`(i>n, [],
      [map(x->[i, x[]], b(n-i, i+1))[], b(n, i+1)[]]))
    end:
T:= n-> map(x-> x[], b(n, 1))[]:
seq(T(n), n=1..12);

Mathematica

T[n_] := Module[{ip, lg}, ip = Reverse /@ Select[ IntegerPartitions[n], # == DeleteDuplicates[#]&]; lg = Length /@ ip // Max; SortBy[PadRight[#, lg]&][ip]];
Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Oct 21 2022 *)