A373953 -id:A373953 - cp's OEIS frontend

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

Gus Wiseman, Jul 19 2024

nonn

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.

			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.

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

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.

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

A374630 Sum of leaders of weakly increasing runs in the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jul 20 2024

nonn

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.

			The maximal weakly increasing subsequences of the 1234567th composition in standard order are ((3),(2),(1,2,2),(1,2,5),(1,1,1)), so a(1234567) = 8.

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

For length instead of sum we have A124766.
For leaders of constant runs we have A373953, excess A373954.
For leaders of anti-runs we have A374516.
Row-sums of A374629.
Counting compositions by this statistic gives A374637.
For leaders of strictly increasing runs we have A374684.
For leaders of weakly decreasing runs we have A374741.
For leaders of strictly decreasing runs we have A374758
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
A373949 counts compositions by run-compressed sum, opposite A373951.
All of the following pertain to compositions in standard order:
- Ones are counted by A000120.
- Sum is A029837 (or sometimes A070939).
- Listed by A066099.
- Length is A070939.
- 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, A374631, A374633, A374635.

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

A374757 Irregular triangle read by rows where row n lists the leaders of strictly decreasing runs in the n-th composition in standard order.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 2, 1, 1, 1, 4, 3, 2, 2, 2, 1, 1, 3, 1, 2, 1, 1, 2, 1, 1, 1, 1, 5, 4, 3, 3, 1, 2, 3, 2, 2, 2, 2, 2, 1, 1, 1, 4, 1, 3, 1, 2, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 6, 5, 4, 4, 1, 3, 3, 3, 3, 2, 3, 1, 1, 2, 4, 2, 3

Offset: 0

Gus Wiseman, Jul 29 2024

The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly 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.

			the 1234567th composition in standard order is (3,2,1,2,2,1,2,5,1,1,1), with strictly decreasing runs ((3,2,1),(2),(2,1),(2),(5,1),(1),(1)), so row 1234567 is (3,2,2,2,5,1,1).
The nonnegative integers, corresponding compositions, and leaders of strictly decreasing runs begin:
    0:      () -> ()        15: (1,1,1,1) -> (1,1,1,1)
    1:     (1) -> (1)       16:       (5) -> (5)
    2:     (2) -> (2)       17:     (4,1) -> (4)
    3:   (1,1) -> (1,1)     18:     (3,2) -> (3)
    4:     (3) -> (3)       19:   (3,1,1) -> (3,1)
    5:   (2,1) -> (2)       20:     (2,3) -> (2,3)
    6:   (1,2) -> (1,2)     21:   (2,2,1) -> (2,2)
    7: (1,1,1) -> (1,1,1)   22:   (2,1,2) -> (2,2)
    8:     (4) -> (4)       23: (2,1,1,1) -> (2,1,1)
    9:   (3,1) -> (3)       24:     (1,4) -> (1,4)
   10:   (2,2) -> (2,2)     25:   (1,3,1) -> (1,3)
   11: (2,1,1) -> (2,1)     26:   (1,2,2) -> (1,2,2)
   12:   (1,3) -> (1,3)     27: (1,2,1,1) -> (1,2,1)
   13: (1,2,1) -> (1,2)     28:   (1,1,3) -> (1,1,3)
   14: (1,1,2) -> (1,1,2)   29: (1,1,2,1) -> (1,1,2)

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

Row-leaders of nonempty rows are A065120.
Row-lengths are A124769.
The opposite version is A374683, sum A374684, length A124768.
The weak version is A374740, sum A374741, length A124765.
Row-sums are A374758.
Positions of identical rows are A374759 (counted by A374760).
Positions of distinct (strict) rows are A374767 (counted by A374761).
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Run-length transform is A333627, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
- Ranks of non-contiguous compositions are A374253, counted by A335548.
Six types of runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A051903, A106356, A188920, A189076, A233564, A238343, A333213, A373949, A374685, A374698, A374700, A374706.

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

A374698 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are distinct.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 27 2024

nonn

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.

			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)

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

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.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],UnsameQ@@First/@Split[stc[#],Less]&]

A374767 Numbers k such that the leaders of strictly decreasing runs in the k-th composition in standard order are distinct.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 24, 25, 32, 33, 34, 35, 37, 38, 40, 41, 44, 48, 49, 50, 52, 64, 65, 66, 67, 68, 69, 70, 72, 74, 75, 77, 78, 80, 81, 82, 83, 88, 89, 92, 96, 97, 98, 101, 102, 104, 105, 108, 128, 129, 130, 131, 132, 133

Offset: 1

Gus Wiseman, Jul 29 2024

nonn

The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly 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.

			The 10000000th composition in standard order is (3,1,4,3,2,1,2,8), with strictly decreasing runs ((3,1),(4,3,2,1),(2),(8)), with leaders (3,4,2,1) so 10000000 is in the sequence.
The terms together with the corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   4: (3)
   5: (2,1)
   6: (1,2)
   8: (4)
   9: (3,1)
  11: (2,1,1)
  12: (1,3)
  13: (1,2,1)
  16: (5)
  17: (4,1)
  18: (3,2)
  19: (3,1,1)
  20: (2,3)
  24: (1,4)
  25: (1,3,1)

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

The opposite version is A374698, counted by A374687.
The weak version is A374701, counted by A374743.
For identical instead of distinct runs we have A374759, counted by A374760.
Compositions of this type are counted by A374761.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Run-length transform is A333627, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
Six types of runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A065120, A106356, A188920, A233564, A238343, A272919, A333213, A373949, A374633, A374685, A374744, A374758, A375128.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],UnsameQ@@First/@Split[stc[#],Greater]&]

A374638 Numbers k such that the leaders of anti-runs in the k-th composition in standard order (A066099) are distinct.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 22, 24, 25, 26, 32, 33, 34, 35, 37, 38, 40, 41, 44, 45, 46, 48, 49, 50, 52, 53, 54, 64, 65, 66, 67, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 83, 88, 89, 91, 92, 93, 96, 97, 98, 100, 101, 102, 104

Offset: 1

Gus Wiseman, Aug 01 2024

nonn

The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) 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.

			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)
  11: (2,1,1)
  12: (1,3)
  13: (1,2,1)
  16: (5)
  17: (4,1)
  18: (3,2)
  19: (3,1,1)
  20: (2,3)
  22: (2,1,2)
  24: (1,4)
  25: (1,3,1)
  26: (1,2,2)

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

Positions of distinct (strict) rows in A374515.
Compositions of this type are counted by A374518.
For identical instead of distinct we have A374519, counted by A374517.
The complement is A374639.
Other types of runs (instead of anti-):
- For identical runs we have A374249, counted by A274174.
- For weakly increasing runs we have A374768, counted by A374632.
- For strictly increasing runs we have A374698, counted by A374687.
- For weakly decreasing runs we have A374701, counted by A374743.
- For strictly decreasing runs we have A374767, counted by A374761.
A065120 gives leaders of standard compositions.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
A238424 counts partitions whose first differences are an anti-run.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Anti-runs are ranked by A333489, counted by A003242.
- Run-length transform is A333627, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
Six types of maximal runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A029931, A114994, A228351, A233564, A238343, A272919, A335467, A373949.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],UnsameQ@@First/@Split[stc[#],UnsameQ]&]

A374685 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are identical.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 10, 12, 13, 14, 15, 16, 20, 24, 25, 27, 28, 29, 30, 31, 32, 36, 40, 42, 48, 49, 51, 52, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 72, 80, 82, 84, 96, 97, 99, 102, 103, 104, 105, 108, 109, 110, 111, 112, 113, 115, 116, 118, 119, 120, 121

Offset: 1

Gus Wiseman, Jul 27 2024

nonn

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.

			The maximal strictly increasing subsequences of the 6560th composition in standard order are ((1,3),(1,2,6)), with leaders (1,1), so 6560 is in the sequence.
The terms together with corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   6: (1,2)
   7: (1,1,1)
   8: (4)
  10: (2,2)
  12: (1,3)
  13: (1,2,1)
  14: (1,1,2)
  15: (1,1,1,1)
  16: (5)
  20: (2,3)
  24: (1,4)
  25: (1,3,1)
  27: (1,2,1,1)
  28: (1,1,3)
  29: (1,1,2,1)
  30: (1,1,1,2)
  31: (1,1,1,1,1)

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

The weak version is A374633, counted by A374631.
Positions of constant rows in A374683.
Compositions of this type are counted by A374686.
For distinct leaders we have A374698, counted by A374687.
The opposite version is A374759, counted by A374760.
Other types of runs: A272919 (counts A000005), A374519 (counts A374517), A374744 (counts A374742).
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A374748 counts compositions by sum of leaders of weakly decreasing 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.
- Ranks of contiguous compositions are A374249, counted by A274174.
Cf. A065120, A106356, A238343, A333213, A373949, A374520, A374629, A374630, A374701, A374740, A374768.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],SameQ@@First/@Split[stc[#],Less]&]

A374519 Numbers k such that the leaders of anti-runs in the k-th composition in standard order (A066099) are identical.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 40, 41, 42, 44, 45, 48, 49, 50, 51, 52, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 72, 73, 76, 77, 80, 81, 82, 84, 85

Offset: 1

Gus Wiseman, Aug 01 2024

nonn

The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) 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.

			The 346th composition in standard order is (2,2,1,2,2), with anti-runs ((2),(2,1,2),(2)), with leaders (2,2,2), so 346 is in the sequence.

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

Positions of constant rows in A374515.
Compositions of this type are counted by A374517.
The complement is A374520.
For distinct instead of identical leaders we have A374638, counted by A374518.
Other types of runs (instead of anti-):
- For identical runs we have A272919, counted by A000005.
- For weakly increasing runs we have A374633, counted by A374631.
- For strictly increasing runs we have A374685, counted by A374686.
- For weakly decreasing runs we have A374744, counted by A374742.
- For strictly decreasing runs we have A374759, counted by A374760.
A065120 gives leaders of standard compositions.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs.
A238424 counts partitions whose first differences are an anti-run.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Anti-runs are ranked by A333489, counted by A003242.
- Run-length transform is A333627, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
Six types of maximal runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A029931, A114994, A228351, A233564, A238343, A373949.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],SameQ@@First/@Split[stc[#],UnsameQ]&]

A374633 Numbers k such that the leaders of weakly increasing runs in the k-th composition in standard order (A066099) are identical.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 10, 12, 13, 14, 15, 16, 20, 24, 25, 26, 27, 28, 29, 30, 31, 32, 36, 40, 42, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 72, 80, 82, 84, 96, 97, 99, 100, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 115

Offset: 1

Gus Wiseman, Jul 21 2024

nonn

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.

			The maximal weakly increasing subsequences of the 26165th composition in standard order are ((1,3),(1,4),(1,2,2),(1)), with leaders (1,1,1,1), so 26165 is in the sequence.
The sequence together with the corresponding compositions begins:
   0: ()
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   6: (1,2)
   7: (1,1,1)
   8: (4)
  10: (2,2)
  12: (1,3)
  13: (1,2,1)
  14: (1,1,2)
  15: (1,1,1,1)
  16: (5)
  20: (2,3)
  24: (1,4)
  25: (1,3,1)
  26: (1,2,2)
  27: (1,2,1,1)

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

For strictly decreasing leaders we appear to have A188920.
For weakly decreasing leaders we appear to have A189076.
Other types of runs: A272919 (counted by A000005), A374519 (counted by A374517), A374685 (counted by A374686), A374744 (counted by A374742), A374759 (counted by A374760).
Positions of constant rows in A374629 (which has sums A374630).
Compositions of this type are counted by A374631.
For strictly increasing leaders see A374634.
For all different leaders we have A374768, counted by A374632.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A374637 counts compositions by sum of leaders of weakly increasing 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 anti-run compositions are A333489, counted by A003242.
- Run-length transform is A333627.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
Cf. A065120, A106356, A238343, A333175, A333213, A373949, A374635.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],SameQ@@First/@Split[stc[#],LessEqual]&]

A374701 Numbers k such that the leaders of weakly decreasing runs in the k-th composition in standard order (A066099) are distinct.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Gus Wiseman, Jul 24 2024

nonn

First differs from A335469 in having 150, which corresponds to the composition (3,2,1,2).

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.

			The maximal weakly decreasing subsequences of the 1257th composition in standard order are ((3,1,1),(2),(3,1)), with leaders (3,2,3), so 1257 is not in the sequence.

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

Positions of distinct (strict) rows in A374740, opposite A374629.
Compositions of this type are counted by A374743.
For identical leaders we have A374744, counted by A374742.
Other types of runs and their counts: A374249 (A274174), A374638 (A374518), A374698 (A374687), A374767 (A374761), A374768 (A374632).
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
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.
- 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, A189076, A238343, A261982, A272919, A333213, A374630, A374635, A374637, A374748.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],UnsameQ@@First/@Split[stc[#],GreaterEqual]&] (* Gus Wiseman, Jul 24 2024 *)

cp's OEIS Frontend

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

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A374630 Sum of leaders of weakly increasing runs in the n-th composition in standard order.

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A374757 Irregular triangle read by rows where row n lists the leaders of strictly decreasing runs in the n-th composition in standard order.

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A374698 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are distinct.

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A374767 Numbers k such that the leaders of strictly decreasing runs in the k-th composition in standard order are distinct.

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A374638 Numbers k such that the leaders of anti-runs in the k-th composition in standard order (A066099) are distinct.

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A374685 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are identical.

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A374519 Numbers k such that the leaders of anti-runs in the k-th composition in standard order (A066099) are identical.

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