A374698 - OEIS

A374698

Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are distinct.

0, 1, 2, 4, 5, 6, 8, 9, 12, 16, 17, 18, 20, 22, 24, 26, 32, 33, 34, 37, 38, 40, 41, 44, 48, 50, 52, 64, 65, 66, 68, 69, 70, 72, 76, 80, 81, 88, 96, 98, 100, 104, 128, 129, 130, 132, 133, 134, 137, 140, 144, 145, 148, 150, 152, 154, 160, 161, 164, 166, 176, 180

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OFFSET

1,3

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.

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..62.

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

EXAMPLE

The maximal strictly increasing subsequences of the 212th composition in standard order are ((1,2),(2,3)), with leaders (1,2), so 212 is in the sequence.

The terms together with corresponding compositions begin:

0: ()

1: (1)

2: (2)

4: (3)

5: (2,1)

6: (1,2)

8: (4)

9: (3,1)

12: (1,3)

16: (5)

17: (4,1)

18: (3,2)

20: (2,3)

22: (2,1,2)

24: (1,4)

26: (1,2,2)

MATHEMATICA

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;

Select[Range[0, 100], UnsameQ@@First/@Split[stc[#], Less]&]

CROSSREFS

Positions of distinct (strict) rows in A374683.

For identical leaders we have A374685, counted by A374761.

Compositions of this type are counted by A374687.

The opposite version is A374767, counted by A374760.

The weak version is A374768, counted by A374632.

Other types of runs: A374249 (counts A274174), A374638 (counts A374518), A374701 (counts A374743).

A011782 counts compositions.

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

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.

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

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

- 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, A106356, A238343, A333213, A373949, A374520, A374629, A374630, A374635, A374740.

Sequence in context: A345167 A370607 A285035 * A233564 A333222 A030326

Adjacent sequences: A374695 A374696 A374697 * A374699 A374700 A374701

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 27 2024

STATUS

approved