A358134 Triangle read by rows whose n-th row lists the partial sums of the n-th composition in standard order (row n of A066099).

1, 2, 1, 2, 3, 2, 3, 1, 3, 1, 2, 3, 4, 3, 4, 2, 4, 2, 3, 4, 1, 4, 1, 3, 4, 1, 2, 4, 1, 2, 3, 4, 5, 4, 5, 3, 5, 3, 4, 5, 2, 5, 2, 4, 5, 2, 3, 5, 2, 3, 4, 5, 1, 5, 1, 4, 5, 1, 3, 5, 1, 3, 4, 5, 1, 2, 5, 1, 2, 4, 5, 1, 2, 3, 5, 1, 2, 3, 4, 5, 6, 5, 6, 4, 6, 4, 5

Offset: 1

Author

Gus Wiseman, Oct 31 2022

Keywords

• nonn
• tabf

Comments

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

Examples

			Triangle begins:
  1
  2
  1 2
  3
  2 3
  1 3
  1 2 3
  4
  3 4
  2 4
  2 3 4
  1 4
  1 3 4
  1 2 4
  1 2 3 4
		

Links

• Gus Wiseman, Statistics, classes, and transformations of standard compositions

Crossrefs

See link for sequences related to standard compositions.

First element in each row is A065120.

Rows are the partial sums of rows of A066099.

Last element in each row is A070939.

An adjusted version is A242628, ranked by A253565.

The first differences instead of partial sums are A358133.

The version for Heinz numbers of partitions is A358136, ranked by A358137.

Row sums are A359042.

A011782 counts compositions.

A351014 counts distinct runs in standard compositions.

A358135 gives last minus first of standard compositions.

Cf. A000120, A001511, A029837, A029931, A048896, A070939, A133494, A253566, A355536, A357135.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Join@@Table[Accumulate[stc[n]],{n,100}]