A389733 Irregular triangle read by rows where row n is the 0-prepended first differences of the n-th composition in standard order.

1, 2, 1, 0, 3, 2, -1, 1, 1, 1, 0, 0, 4, 3, -2, 2, 0, 2, -1, 0, 1, 2, 1, 1, -1, 1, 0, 1, 1, 0, 0, 0, 5, 4, -3, 3, -1, 3, -2, 0, 2, 1, 2, 0, -1, 2, -1, 1, 2, -1, 0, 0, 1, 3, 1, 2, -2, 1, 1, 0, 1, 1, -1, 0, 1, 0, 2, 1, 0, 1, -1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 6, 5, -4, 4, -2

Offset

1,2

Comments

Row 0 is empty so offset is 1.

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=1..85.

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

Example

The 6th standard composition is (1,2), with 0-prepended parts (0,1,2), with first differences (1,1), so row 6 is (1,1).
Triangle begins:
   0: .
   1: 1
   2: 2
   3: 1  0
   4: 3
   5: 2 -1
   6: 1  1
   7: 1  0  0
   8: 4
   9: 3 -2
  10: 2  0
  11: 2 -1  0
  12: 1  2
  13: 1  1 -1
  14: 1  0  1
  15: 1  0  0  0
  16: 5
  17: 4 -3
  18: 3 -1
  19: 3 -2  0
  20: 2  1

Mathematica

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Differences[Prepend[stc[n], 0]], {n, 0, 10}]

Crossrefs

Row lengths are A000120.

Row sums are A001511.

First term in each row is A065120.

Zeros in each row are counted by A124762.

For Heinz numbers of partitions we get A287352, without prepending A355536.

Without prepending we get A358133.

Positions of constant rows are A389732.

Positions of strict rows are A389734, without prepending A389597.

A007862 counts partitions with constant 0-prepended differences, ranks A325327.

A011782 counts compositions.

A066099 lists compositions in standard order.

A070939 gives sum of standard compositions.

A124767 counts maximal runs in standard compositions, anti-runs A333381.

Cf. A029837, A029931, A048896, A175342, A242628, A351014, A358134, A358135.

Sequence in context: A274183 A212278 A244215 * A391983 A255325 A391368

Adjacent sequences: A389730 A389731 A389732 * A389734 A389735 A389736

Keyword

sign,tabf

Author

Gus Wiseman, Oct 23 2025

Status

approved