A357187 First differences A357186 = "Take the k-th composition in standard order for each part k of the n-th composition in standard order, then add up everything."

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

Offset

0,32

Comments

Are there any terms > 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=0..81.

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

Formula

a(n) = A357186(n + 1) - A357186(n).

Example

We have A357186(5) - A357186(4) = 3 - 2 = 1, so a(4) = 1.

Mathematica

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

Crossrefs

See link for sequences related to standard compositions.

Positions of first appearances appear to all belong to A052955.

Differences of A357186 (row-sums of A357135).

The version for partitions is A357458, differences of A325033.

Cf. A000120, A029837, A029931, A048896, A058891, A070939, A133494, A333766, A357134, A357137.

Sequence in context: A090584 A171400 A271592 * A128409 A133699 A157361

Adjacent sequences: A357184 A357185 A357186 * A357188 A357189 A357190

Keyword

sign

Author

Gus Wiseman, Sep 28 2022

Status

approved