A374741 Sum of leaders of weakly decreasing runs in the n-th composition in standard order.

0, 1, 2, 1, 3, 2, 3, 1, 4, 3, 2, 2, 4, 3, 3, 1, 5, 4, 3, 3, 5, 2, 4, 2, 5, 4, 3, 3, 4, 3, 3, 1, 6, 5, 4, 4, 3, 3, 5, 3, 6, 5, 2, 2, 5, 4, 4, 2, 6, 5, 4, 4, 6, 3, 5, 3, 5, 4, 3, 3, 4, 3, 3, 1, 7, 6, 5, 5, 4, 4, 6, 4, 7, 3, 3, 3, 6, 5, 5, 3, 7, 6, 5, 5, 5, 2, 4

Offset

0,3

Comments

The leaders of weakly decreasing runs in a sequence are obtained by splitting it into maximal weakly decreasing subsequences and taking the first term of each.

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, Sequences counting and ranking compositions by their leaders (for six types of runs).

Example

The maximal weakly decreasing subsequences of the 1234567th composition in standard order are ((3,2,1),(2,2,1),(2),(5,1,1,1)), so a(1234567) is 3+2+2+5 = 12.

Mathematica

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

Crossrefs

For length instead of sum we have A124765.

Other types of runs are A373953, A374516, A374684, A374758.

The opposite is A374630.

Row-sums of A374740, opposite A374629.

Counting compositions by this statistic gives A374748, opposite A374637.

A373949 counts compositions by run-compressed sum.

All of the following pertain to compositions in standard order:

- Length is A000120.

- Sum is A029837(n+1) (or sometimes A070939).

- Parts are listed by A066099.

- Number of adjacent equal pairs is A124762, unequal A333382.

- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.

- Ranks of strict compositions are A233564, counted by A032020.

- Constant compositions are ranked by A272919.

- Ranks of anti-run compositions are A333489, counted by A003242.

- Run-length transform is A333627.

- Run-compression transform is A373948.

Cf. A065120, A106356, A188920, A189076, A238343, A333175, A333213, A373949, A373954, A374631, A374633, A374635, A374701, A374742.

Sequence in context: A097285 A372917 A057432 * A361942 A302295 A215467

Adjacent sequences: A374738 A374739 A374740 * A374742 A374743 A374744

Keyword

nonn,new

Author

Gus Wiseman, Jul 24 2024

Status

approved