A374684 Sum of leaders of strictly increasing runs in the n-th composition in standard order.

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

Offset

0,3

Comments

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

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 strictly increasing subsequences of the 1234567th composition in standard order are ((3),(2),(1,2),(2),(1,2,5),(1),(1),(1)) with leaders (3,2,1,2,1,1,1,1), so a(1234567) = 12.

Mathematica

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

Crossrefs

The weak version is A374630.

Row-sums of A374683.

The opposite version is A374758.

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.

- Run-length transform is A333627.

- Run-compression transform is A373948.

- Ranks of contiguous compositions are A374249, counted by A274174.

- Ranks of non-contiguous compositions are A374253, counted by A335548.

Cf. A065120, A106356, A188920, A189076, A238343, A272919, A374634, A374685, A374698.

Cf. A374251 (sums A373953), A374515 (sums A374516), A374740 (sums A374741).

Sequence in context: A128924 A364569 A239957 * A230040 A242361 A116464

Adjacent sequences: A374681 A374682 A374683 * A374685 A374686 A374687

Keyword

nonn

Author

Gus Wiseman, Jul 26 2024

Status

approved