A358133 Triangle read by rows whose n-th row lists the first differences of the n-th composition in standard order (row n of A066099).

0, -1, 1, 0, 0, -2, 0, -1, 0, 2, 1, -1, 0, 1, 0, 0, 0, -3, -1, -2, 0, 1, 0, -1, -1, 1, -1, 0, 0, 3, 2, -2, 1, 0, 1, -1, 0, 0, 2, 0, 1, -1, 0, 0, 1, 0, 0, 0, 0, -4, -2, -3, 0, 0, -1, -1, -2, 1, -2, 0, 0, 2, 1, -2, 0, 0, 0, -1, 0, -1, 2, -1, 1, -1, -1, 0, 1, -1

Offset

3,6

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.

Links

Table of n, a(n) for n=3..80.

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

Example

Triangle begins (dots indicate empty rows):
   1:   .
   2:   .
   3:   0
   4:   .
   5:  -1
   6:   1
   7:   0  0
   8:   .
   9:  -2
  10:   0
  11:  -1  0
  12:   2
  13:   1 -1
  14:   0  1
  15:   0  0  0

Mathematica

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

Crossrefs

See link for sequences related to standard compositions.

First differences of rows of A066099.

The version for Heinz numbers of partitions is A355536, ranked by A253566.

The partial sums instead of first differences are A358134.

Row sums are A358135.

A011782 counts compositions.

A351014 counts distinct runs in standard compositions.

Cf. A000120, A001511, A029837, A029931, A048896, A070939, A133494, A242628, A357135, A357187.

Sequence in context: A127505 A194923 A321103 * A363878 A086372 A342003

Adjacent sequences: A358130 A358131 A358132 * A358134 A358135 A358136

Keyword

sign,tabf

Author

Gus Wiseman, Oct 31 2022

Status

approved