A124769 -id:A124769 - cp's OEIS frontend

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

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 *)

A375123 Weakly increasing run-leader transformation for standard compositions.

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 1, 1, 8, 9, 2, 5, 1, 3, 1, 1, 16, 17, 18, 9, 2, 5, 5, 5, 1, 3, 1, 3, 1, 3, 1, 1, 32, 33, 34, 17, 4, 37, 9, 9, 2, 5, 2, 5, 5, 11, 5, 5, 1, 3, 6, 3, 1, 3, 3, 3, 1, 3, 1, 3, 1, 3, 1, 1, 64, 65, 66, 33, 68, 69, 17, 17, 4, 9, 18, 37, 9, 19, 9, 9

Offset: 0

Gus Wiseman, Aug 02 2024

nonn

The a(n)-th composition in standard order lists the leaders of weakly increasing runs of the n-th composition in standard order.
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 813th composition in standard order is (1,3,2,1,2,1), with weakly increasing runs ((1,3),(2),(1,2),(1)), with leaders (1,2,1,1). This is the 27th composition in standard order, so a(813) = 27.

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

Positions of elements of A233564 are A374768, counted by A374632.
Positions of elements of A272919 are A374633, counted by A374631.
Ranks of rows of A374629.
The opposite version is A375124.
The strict version is A375125.
The strict opposite version is A375126.
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).
- Leader is A065120.
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Run-length transform is A333627, sum A070939.
- Run-sum transformation is A353847.
- 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. A189076, A238343, A333213, A373949, A374634, A374635, A374637, A374743, A374744, A375128.

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

A000120(a(n)) = A124766(n).
A070939(a(n)) = A374630(n) for n > 0.
A065120(a(n)) = A065120(n).

A374759 Numbers k such that the leaders of strictly decreasing runs in the k-th composition in standard order are identical.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 15, 16, 17, 18, 21, 22, 31, 32, 33, 34, 36, 37, 42, 45, 63, 64, 65, 66, 68, 69, 73, 76, 85, 86, 90, 127, 128, 129, 130, 132, 133, 136, 137, 146, 148, 153, 170, 173, 181, 182

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 18789th composition in standard order is (3,3,2,1,3,2,1), with strictly decreasing runs ((3),(3,2,1),(3,2,1)), with leaders (3,3,3), so 18789 is in the sequence.
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)
  15: (1,1,1,1)
  16: (5)
  17: (4,1)
  18: (3,2)
  21: (2,2,1)
  22: (2,1,2)
  31: (1,1,1,1,1)
  32: (6)
  33: (5,1)
  34: (4,2)
  36: (3,3)
  37: (3,2,1)

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

For leaders of anti-runs we have A374519 (counted by A374517).
For leaders of weakly increasing runs we have A374633, counted by A374631.
The opposite version is A374685 (counted by A374686).
The weak version is A374744.
Compositions of this type are counted by A374760.
For distinct instead of identical runs we have 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. A000961, A065120, A106356, A188920, A189076, A238343, A272919, A333213, A374698, A374706, A374758, A375128.

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

A333230 Positions of weak ascents in the sequence of differences between primes.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 10, 13, 14, 15, 17, 20, 22, 23, 26, 28, 29, 31, 33, 35, 36, 38, 39, 41, 43, 45, 46, 49, 50, 52, 54, 55, 57, 60, 61, 64, 65, 67, 69, 70, 71, 73, 75, 76, 78, 79, 81, 83, 85, 86, 89, 90, 93, 95, 96, 98, 100, 102, 104, 105, 107, 109, 110, 113

Offset: 1

Gus Wiseman, Mar 18 2020

Partial sums of A333252.

			The prime gaps split into the following strictly decreasing subsequences: (1), (2), (2), (4,2), (4,2), (4), (6,2), (6,4,2), (4), (6), (6,2), (6,4,2), (6,4), (6), (8,4,2), ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Thomas Bloom, Let d_n = p_(n+1)-p_n. The set of n such that d_(n+1) >= d_n has density 1/2, and similarly for d_(n+1) <= d_n. Furthermore, there are infinitely many n such that d_(n+1) = d_n, Erdős Problems.
Terence Tao, Erdős problem database, see no. 218.

The version for the Kolakoski sequence is A022297.
The version for equal differences is A064113.
The version for strict ascents is A258025.
The version for strict descents is A258026.
The version for distinct differences is A333214.
The version for weak descents is A333231.
First differences are A333252 (if the first term is 0).
Prime gaps are A001223.
Weakly decreasing runs of standard compositions are counted by A124765.
Weakly increasing runs of standard compositions are counted by A124766.
Strictly increasing runs of standard compositions are counted by A124768.
Strictly decreasing runs of standard compositions are counted by A124769.
Runs of prime gaps with nonzero differences are A333216.
Cf. A000040, A000720, A036263, A054819, A084758, A333212, A333213, A333215.

Accumulate[Length/@Split[Differences[Array[Prime,100]],#1>#2&]]//Most
(* or *)
Select[Range[100],Prime[#+1]-Prime[#]<=Prime[#+2]-Prime[#+1]&]

Numbers k such that prime(k+2) - 2*prime(k+1) + prime(k) >= 0.

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]&]

A375124 Weakly decreasing run-leader transformation for standard compositions.

Original entry on oeis.org

0, 1, 2, 1, 4, 2, 6, 1, 8, 4, 2, 2, 12, 6, 6, 1, 16, 8, 4, 4, 20, 2, 10, 2, 24, 12, 6, 6, 12, 6, 6, 1, 32, 16, 8, 8, 4, 4, 18, 4, 40, 20, 2, 2, 20, 10, 10, 2, 48, 24, 12, 12, 52, 6, 26, 6, 24, 12, 6, 6, 12, 6, 6, 1, 64, 32, 16, 16, 8, 8, 34, 8, 72, 4, 4, 4, 36

Offset: 0

Gus Wiseman, Aug 02 2024

nonn

The a(n)-th composition in standard order lists the leaders of weakly decreasing runs in the n-th composition in standard order.
The leaders of weakly decreasing runs in a sequence are obtained by splitting it 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 813th composition in standard order is (1,3,2,1,2,1), with weakly decreasing runs ((1),(3,2,1),(2,1)), with leaders (1,3,2). This is the 50th composition in standard order, so a(813) = 50.

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

Positions of elements of A233564 are A374701, counted by A374743.
Positions of elements of A272919 are A374744, counted by A374742.
Ranks of rows of A374740.
The opposite version is A375123.
The strict version is A375126.
The strict opposite version is A375125.
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) = A070939(n).
- 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.
- Run-sum transformation is A353847.
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, A189076, A238343, A333213, A373949, A374635, A374687, A374748, A374758.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[First/@Split[stc[n],GreaterEqual]],{n,0,100}]

A000120(a(n)) = A124765(n).
A065120(a(n)) = A065120(n).
A070939(a(n)) = A374741(n).

A375125 Strictly increasing run-leader transformation for standard compositions.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 1, 7, 8, 9, 10, 11, 1, 3, 3, 15, 16, 17, 18, 19, 2, 21, 5, 23, 1, 3, 6, 7, 3, 7, 7, 31, 32, 33, 34, 35, 36, 37, 9, 39, 2, 5, 42, 43, 5, 11, 11, 47, 1, 3, 6, 7, 1, 13, 3, 15, 3, 7, 14, 15, 7, 15, 15, 63, 64, 65, 66, 67, 68, 69, 17, 71, 4, 73

Offset: 0

Gus Wiseman, Aug 02 2024

nonn

The a(n)-th composition in standard order lists the leaders of strictly increasing runs in the n-th composition in standard order.
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 813th composition in standard order is (1,3,2,1,2,1), with strictly increasing runs ((1,3),(2),(1,2),(1)), with leaders (1,2,1,1). This is the 27th composition in standard order, so a(813) = 27.

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

Positions of elements of A233564 are A374698, counted by A374687.
Positions of elements of A272919 are A374685, counted by A374686.
Ranks of rows of A374683.
The weak version is A375123.
The weak opposite version is A375124.
The opposite version is A375126.
Other transformations: A375127, A373948.
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) = A070939(n).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Run-length transform is A333627.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
- Run-sum transformation is A353847.
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, A189076, A238343, A333213, A373949, A374634, A374635, A374684, A374700, A375128.

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

A000120(a(n)) = A124768(n).
A065120(a(n)) = A065120(n).
A070939(a(n)) = A374684(n).

cp's OEIS Frontend

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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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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A375123 Weakly increasing run-leader transformation for standard compositions.

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A374759 Numbers k such that the leaders of strictly decreasing runs in the k-th composition in standard order are identical.

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A333230 Positions of weak ascents in the sequence of differences between primes.

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

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