A066099 -id:A066099 - cp's OEIS frontend

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

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]&]

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 *)

A334968 Number of possible sums of subsequences (not necessarily contiguous) of the n-th composition in standard order (A066099).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 4, 4, 2, 4, 3, 5, 4, 5, 5, 5, 2, 4, 4, 6, 4, 6, 6, 6, 4, 6, 6, 6, 6, 6, 6, 6, 2, 4, 4, 6, 3, 7, 7, 7, 4, 7, 4, 7, 7, 7, 7, 7, 4, 6, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 2, 4, 4, 6, 4, 8, 8, 8, 4, 6, 6, 8, 6, 8, 8, 8, 4, 8, 6, 8, 6, 8, 8

Offset: 0

Gus Wiseman, Jun 02 2020

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 139th composition is (4,2,1,1), with possible sums of subsequences {0,1,2,3,4,5,6,7,8}, so a(139) = 9.
Triangle begins:
  1
  2
  2 3
  2 4 4 4
  2 4 3 5 4 5 5 5
  2 4 4 6 4 6 6 6 4 6 6 6 6 6 6 6
  2 4 4 6 3 7 7 7 4 7 4 7 7 7 7 7 4 6 7 7 7 7 7 7 6 7 7 7 7 7 7 7

Row lengths are A011782.
Dominated by A124771 (number of contiguous subsequences).
Dominates A333257 (the contiguous case).
Dominated by A334299 (number of subsequences).
Golomb rulers are counted by A169942 and ranked by A333222.
Positive subset-sums of partitions are counted by A276024 and A299701.
Knapsack partitions are counted by A108917 and ranked by A299702
Knapsack compositions are counted by A325676 and ranked by A333223.
Contiguous subsequence-sums are counted by A333224 and ranked by A333257.
Knapsack compositions are counted by A334268 and ranked by A334967.
Cf. A000120, A029931, A048793, A066099, A070939, A108917, A325769, A325770, A325778, A334300, A335279.

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

a(n) = A299701(A333219(n)).

A359042 Sum of partial sums of the n-th composition in standard order (A066099).

Original entry on oeis.org

0, 1, 2, 3, 3, 5, 4, 6, 4, 7, 6, 9, 5, 8, 7, 10, 5, 9, 8, 12, 7, 11, 10, 14, 6, 10, 9, 13, 8, 12, 11, 15, 6, 11, 10, 15, 9, 14, 13, 18, 8, 13, 12, 17, 11, 16, 15, 20, 7, 12, 11, 16, 10, 15, 14, 19, 9, 14, 13, 18, 12, 17, 16, 21, 7, 13, 12, 18, 11, 17, 16, 22

Offset: 0

Gus Wiseman, Dec 20 2022

nonn

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 29th composition in standard order is (1,1,2,1), with partial sums (1,2,4,5), with sum 12, so a(29) = 12.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
Each n appears A000009(n) times.
The reverse version is A029931.
Comps counted by this statistic are A053632, ptns A264034, rev ptns A358194.
This is the sum of partial sums of rows of A066099.
The version for Heinz numbers of partitions is A318283, row sums of A358136.
Row sums of A358134.
A011782 counts compositions.
A065120 gives first part of standard compositions, last A001511.
A242628 lists adjusted partial sums, ranked by A253565, row sums A359043.
A358135 gives last minus first of standard compositions.
Cf. A000120, A029837, A070939, A133494, A253566, A358133, A358137.

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

A359043 Sum of adjusted partial sums of the n-th composition in standard order (A066099). Row sums of A242628.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 3, 3, 4, 6, 5, 6, 4, 5, 4, 4, 5, 8, 7, 9, 6, 8, 7, 8, 5, 7, 6, 7, 5, 6, 5, 5, 6, 10, 9, 12, 8, 11, 10, 12, 7, 10, 9, 11, 8, 10, 9, 10, 6, 9, 8, 10, 7, 9, 8, 9, 6, 8, 7, 8, 6, 7, 6, 6, 7, 12, 11, 15, 10, 14, 13, 16, 9, 13, 12, 15, 11, 14, 13

Offset: 0

Gus Wiseman, Dec 21 2022

nonn

We define the adjusted partial sums of a composition to be obtained by subtracting one from all parts, taking partial sums, and adding one back to all parts.

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 29th composition in standard order is (1,1,2,1), with adjusted partial sums (1,1,2,2), with sum 6, so a(29) = 6.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
The unadjusted reverse version is A029931, row sums of A048793.
The reverse version is A161511, row sums of A125106.
Row sums of A242628, ranked by A253565.
The unadjusted version is A359042, row sums of A358134.
A011782 counts compositions.
A066099 lists standard compositions.
A358135 gives last minus first of standard compositions.
A358194 counts partitions by sum and weighted sum.
Cf. A000120, A005940, A019565, A029837, A059893, A253566, A358133, A358170.

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

A335238 Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are not pairwise coprime, where a singleton is not coprime unless it is (1).

Original entry on oeis.org

0, 2, 4, 8, 10, 16, 32, 34, 36, 40, 42, 64, 69, 70, 81, 88, 98, 104, 128, 130, 136, 138, 139, 141, 142, 160, 162, 163, 168, 170, 177, 184, 197, 198, 209, 216, 226, 232, 256, 260, 261, 262, 274, 276, 277, 278, 279, 282, 283, 285, 286, 288, 290, 292, 296, 321

Offset: 1

Gus Wiseman, May 28 2020

nonn

We use the Mathematica definition for CoprimeQ, so a singleton is not considered coprime unless it is (1).

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 sequence together with the corresponding compositions begins:
    0: ()          88: (2,1,4)      177: (2,1,4,1)
    2: (2)         98: (1,4,2)      184: (2,1,1,4)
    4: (3)        104: (1,2,4)      197: (1,4,2,1)
    8: (4)        128: (8)          198: (1,4,1,2)
   10: (2,2)      130: (6,2)        209: (1,2,4,1)
   16: (5)        136: (4,4)        216: (1,2,1,4)
   32: (6)        138: (4,2,2)      226: (1,1,4,2)
   34: (4,2)      139: (4,2,1,1)    232: (1,1,2,4)
   36: (3,3)      141: (4,1,2,1)    256: (9)
   40: (2,4)      142: (4,1,1,2)    260: (6,3)
   42: (2,2,2)    160: (2,6)        261: (6,2,1)
   64: (7)        162: (2,4,2)      262: (6,1,2)
   69: (4,2,1)    163: (2,4,1,1)    274: (4,3,2)
   70: (4,1,2)    168: (2,2,4)      276: (4,2,3)
   81: (2,4,1)    170: (2,2,2,2)    277: (4,2,2,1)

The complement is A333228.
Not ignoring repeated parts gives A335239.
Singleton or pairwise coprime partitions are counted by A051424.
Singleton or pairwise coprime sets are ranked by A087087.
Coprime partitions are counted by A327516.
Non-coprime partitions are counted by A335240.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Product is A124758.
- Reverse is A228351
- GCD is A326674.
- Heinz number is A333219.
- LCM is A333226.
- Coprime compositions are A333227.
- Compositions whose distinct parts are coprime are A333228.
- Number of distinct parts is A334028.
Cf. A007360, A048793, A101268, A291166, A302569, A326675, A335235, A335236, A335237.

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

A358134 Triangle read by rows whose n-th row lists the partial sums of the n-th composition in standard order (row n of A066099).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 31 2022

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.

			Triangle begins:
  1
  2
  1 2
  3
  2 3
  1 3
  1 2 3
  4
  3 4
  2 4
  2 3 4
  1 4
  1 3 4
  1 2 4
  1 2 3 4

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
First element in each row is A065120.
Rows are the partial sums of rows of A066099.
Last element in each row is A070939.
An adjusted version is A242628, ranked by A253565.
The first differences instead of partial sums are A358133.
The version for Heinz numbers of partitions is A358136, ranked by A358137.
Row sums are A359042.
A011782 counts compositions.
A351014 counts distinct runs in standard compositions.
A358135 gives last minus first of standard compositions.
Cf. A000120, A001511, A029837, A029931, A048896, A070939, A133494, A253566, A355536, A357135.

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

A374744 Numbers k such that the leaders of weakly decreasing runs in the k-th composition in standard order (A066099) are identical.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 15, 16, 17, 18, 19, 21, 22, 23, 31, 32, 33, 34, 35, 36, 37, 39, 42, 43, 45, 46, 47, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 79, 85, 86, 87, 90, 91, 93, 94, 95, 127, 128, 129, 130, 131, 132, 133, 135, 136, 137, 138

Offset: 1

Gus Wiseman, Jul 24 2024

nonn

The leaders of weakly decreasing runs in a sequence are obtained by splitting 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.

			The terms together with the corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   5: (2,1)
   7: (1,1,1)
   8: (4)
   9: (3,1)
  10: (2,2)
  11: (2,1,1)
  15: (1,1,1,1)
  16: (5)
  17: (4,1)
  18: (3,2)
  19: (3,1,1)
  21: (2,2,1)
  22: (2,1,2)
  23: (2,1,1,1)
  31: (1,1,1,1,1)

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

Other types of runs and their counts: A272919 (A000005), A374519 (A374517), A374685 (A374686), A374759 (A374760).
The opposite is A374633, counted by A374631.
For distinct (instead of identical) leaders we have A374701, count A374743.
Positions of constant rows in A374740, opposite A374629, cf. A374630.
Compositions of this type are counted by 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, A333175, A333213, A335456, A373949, A374635, A374634, A374768.

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

cp's OEIS Frontend

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

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

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A334968 Number of possible sums of subsequences (not necessarily contiguous) of the n-th composition in standard order (A066099).

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Formula

A359042 Sum of partial sums of the n-th composition in standard order (A066099).

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A359043 Sum of adjusted partial sums of the n-th composition in standard order (A066099). Row sums of A242628.

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A335238 Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are not pairwise coprime, where a singleton is not coprime unless it is (1).

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