A357135 Take the k-th composition in standard order for each part k of the n-th composition in standard order; then concatenate.

1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1

Offset

0,2

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=0..86.

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

Formula

Row n is the A357134(n)-th composition in standard order.

Example

Triangle begins:
   0:
   1: 1
   2: 2
   3: 1 1
   4: 1 1
   5: 2 1
   6: 1 2
   7: 1 1 1
   8: 3
   9: 1 1 1
  10: 2 2
  11: 2 1 1
  12: 1 1 1
  13: 1 2 1
  14: 1 1 2
  15: 1 1 1 1

Mathematica

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Join@@Table[Join@@stc/@stc[n], {n, 0, 30}]

Crossrefs

See link for sequences related to standard compositions.

Row n is the A357134(n)-th composition in standard order.

The version for Heinz numbers of partitions is A357139, cf. A003963.

Row sums are A357186, differences A357187.

Cf. A000120, A001511, A029931, A048896, A058891, A070939, A096111, A329395, A333766, A335404, A357137.

Sequence in context: A193348 A263723 A354974 * A367987 A370077 A370080

Adjacent sequences: A357132 A357133 A357134 * A357136 A357137 A357138

Keyword

nonn

Author

Gus Wiseman, Sep 26 2022

Status

approved