A374768 Numbers k such that the leaders of weakly increasing runs in the k-th composition in standard order (A066099) are distinct.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 50, 52, 56, 58, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81

Offset

1,3

Comments

First differs from A335467 in having 166, corresponding to the composition (2,3,1,2).

The leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing 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=1..68.

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

Example

The 4444th composition in standard order is (4,2,2,1,1,3), with weakly increasing runs ((4),(2,2),(1,1,3)), with leaders (4,2,1), so 4444 is in the sequence.

Mathematica

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 166], UnsameQ@@First/@Split[stc[#], LessEqual]&]

Crossrefs

These are the positions of strict rows in A374629 (which has sums A374630).

Compositions of this type are counted by A374632, increasing A374634.

Identical instead of distinct leaders are A374633, counted by A374631.

For leaders of anti-runs we have A374638, counted by A374518.

For leaders of strictly increasing runs we have A374698, counted by A374687.

For leaders of weakly decreasing runs we have A374701, counted by A374743.

For leaders of strictly decreasing runs we have A374767, counted by A374761.

A011782 counts compositions.

A238130, A238279, A333755 count compositions by number of runs.

All of the following pertain to compositions in standard order:

- Ones are counted by A000120.

- Sum is A029837 (or sometimes A070939).

- Parts are listed by A066099.

- Length is A070939.

- Adjacent equal pairs are counted by A124762, unequal A333382.

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

- Ranks of strict compositions are A233564.

- Ranks of constant compositions are A272919.

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

- Run-length transform is A333627.

- Run-compression transform is A373948, sum A373953, excess A373954.

Cf. A065120, A188920, A189076, A238343, A333213, A335449, A373949, A274174, A374249, A374635, A374637.

Sequence in context: A023754 A371968 A335467 * A256474 A085824 A351290

Adjacent sequences: A374753 A374755 A374756 * A374770 A374771 A374772

Keyword

nonn,new

Author

Gus Wiseman, Jul 19 2024

Status

approved