A359043 Sum of adjusted partial sums of the n-th composition in standard order (A066099). Row sums of A242628.

0, 1, 2, 2, 3, 4, 3, 3, 4, 6, 5, 6, 4, 5, 4, 4, 5, 8, 7, 9, 6, 8, 7, 8, 5, 7, 6, 7, 5, 6, 5, 5, 6, 10, 9, 12, 8, 11, 10, 12, 7, 10, 9, 11, 8, 10, 9, 10, 6, 9, 8, 10, 7, 9, 8, 9, 6, 8, 7, 8, 6, 7, 6, 6, 7, 12, 11, 15, 10, 14, 13, 16, 9, 13, 12, 15, 11, 14, 13

Offset

0,3

Comments

We define the adjusted partial sums of a composition to be obtained by subtracting one from all parts, taking partial sums, and adding one back to all parts.

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..78.

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

Example

The 29th composition in standard order is (1,1,2,1), with adjusted partial sums (1,1,2,2), with sum 6, so a(29) = 6.

Mathematica

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

Crossrefs

See link for sequences related to standard compositions.

The unadjusted reverse version is A029931, row sums of A048793.

The reverse version is A161511, row sums of A125106.

Row sums of A242628, ranked by A253565.

The unadjusted version is A359042, row sums of A358134.

A011782 counts compositions.

A066099 lists standard compositions.

A358135 gives last minus first of standard compositions.

A358194 counts partitions by sum and weighted sum.

Cf. A000120, A005940, A019565, A029837, A059893, A253566, A358133, A358170.

Sequence in context: A350238 A352746 A330416 * A054703 A048206 A290740

Adjacent sequences: A359040 A359041 A359042 * A359044 A359045 A359046

Keyword

nonn

Author

Gus Wiseman, Dec 21 2022

Status

approved