A318928 -id:A318928 - cp's OEIS frontend

A353402 Numbers k such that the k-th composition in standard order has its own run-lengths as a subsequence (not necessarily consecutive).

0, 1, 10, 21, 26, 43, 53, 58, 107, 117, 174, 186, 292, 314, 346, 348, 349, 373, 430, 442, 570, 585, 586, 629, 676, 693, 696, 697, 698, 699, 804, 826, 858, 860, 861, 885, 954, 1082, 1141, 1173, 1210, 1338, 1353, 1387, 1392, 1393, 1394, 1396, 1397, 1398, 1466

Offset: 0

Gus Wiseman, May 15 2022

nonn

First differs from A353432 (the consecutive case) in having 0 and 53.

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 initial terms, their binary expansions, and the corresponding standard compositions:
    0:          0  ()
    1:          1  (1)
   10:       1010  (2,2)
   21:      10101  (2,2,1)
   26:      11010  (1,2,2)
   43:     101011  (2,2,1,1)
   53:     110101  (1,2,2,1)
   58:     111010  (1,1,2,2)
  107:    1101011  (1,2,2,1,1)
  117:    1110101  (1,1,2,2,1)
  174:   10101110  (2,2,1,1,2)
  186:   10111010  (2,1,1,2,2)
  292:  100100100  (3,3,3)
  314:  100111010  (3,1,1,2,2)
  346:  101011010  (2,2,1,2,2)
  348:  101011100  (2,2,1,1,3)
  349:  101011101  (2,2,1,1,2,1)
  373:  101110101  (2,1,1,2,2,1)
  430:  110101110  (1,2,2,1,1,2)
  442:  110111010  (1,2,1,1,2,2)

The version for partitions is A325755, counted by A325702.
These compositions are counted by A353390.
The recursive version is A353431, counted by A353391.
The consecutive case is A353432, counted by A353392.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order, reverse A228351.
A333769 lists run-lengths of compositions in standard order.
Words with all distinct run-lengths: A032020, A044813, A098859, A130091, A329739, A351017.
Statistics of standard compositions:
- Length is A000120, sum A070939.
- Runs are counted by A124767, distinct A351014.
- Subsequences are counted by A334299, consecutive A124770/A124771.
- Runs-resistance is A333628.
Classes of standard compositions:
- Partitions are A114994, strict A333255, rev A225620, strict rev A333256.
- Runs are A272919.
- Golomb rulers are A333222, counted by A169942.
- Knapsack compositions are A333223, counted by A325676.
- Anti-runs are A333489, counted by A003242.
Cf. A114640, A165413, A181819, A318928, A325705, A329738, A333224/A333257, A333755, A353393, A353403, A353430.

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
rosQ[y_]:=Length[y]==0||MemberQ[Subsets[y],Length/@Split[y]];
Select[Range[0,100],rosQ[stc[#]]&]

A329862 Positive integers whose binary expansion has cuts-resistance 2.

Original entry on oeis.org

3, 4, 6, 9, 11, 12, 13, 18, 19, 20, 22, 25, 26, 37, 38, 41, 43, 44, 45, 50, 51, 52, 53, 74, 75, 76, 77, 82, 83, 84, 86, 89, 90, 101, 102, 105, 106, 149, 150, 153, 154, 165, 166, 169, 171, 172, 173, 178, 179, 180, 181, 202, 203, 204, 205, 210, 211, 212, 213

Offset: 1

Gus Wiseman, Nov 23 2019

nonn

For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.

			The sequence of terms together with their binary expansions begins:
   3:      11
   4:     100
   6:     110
   9:    1001
  11:    1011
  12:    1100
  13:    1101
  18:   10010
  19:   10011
  20:   10100
  22:   10110
  25:   11001
  26:   11010
  37:  100101
  38:  100110
  41:  101001
  43:  101011
  44:  101100
  45:  101101
  50:  110010

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Positions of 2's in A319416.
Numbers whose binary expansion has cuts-resistance 1 are A000975.
Binary words with cuts-resistance 2 are conjectured to be A027383.
Compositions with cuts-resistance 2 are A329863.
Cuts-resistance of binary expansion without first digit is A319420.
Binary words counted by cuts-resistance are A319421 and A329860.
Compositions counted by cuts-resistance are A329861.
Cf. A107907, A114901, A164707, A318928, A319411, A329745, A329865, A329866.

degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Select[Range[100],degdep[IntegerDigits[#,2]]==2&]

A353431 Numbers k such that the k-th composition in standard order is empty, a singleton, or has its own run-lengths as a subsequence (not necessarily consecutive) that is already counted.

Original entry on oeis.org

0, 1, 2, 4, 8, 10, 16, 32, 43, 58, 64, 128, 256, 292, 349, 442, 512, 586, 676, 697, 826, 1024, 1210, 1338, 1393, 1394, 1396, 1594, 2048, 2186, 2234, 2618, 2696, 2785, 2786, 2792, 3130, 4096, 4282, 4410, 4666, 5178, 5569, 5570, 5572, 5576, 5584, 6202, 8192

Offset: 1

Gus Wiseman, May 16 2022

nonn

First differs from A353696 (the consecutive version) in having 22318, corresponding to the binary word 101011100101110 and standard composition (2,2,1,1,3,2,1,1,2), whose run-lengths (2,2,1,1,2,1) are subsequence but not a consecutive subsequence.

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 initial terms, their binary expansions, and the corresponding standard compositions:
     0:           0  ()
     1:           1  (1)
     2:          10  (2)
     4:         100  (3)
     8:        1000  (4)
    10:        1010  (2,2)
    16:       10000  (5)
    32:      100000  (6)
    43:      101011  (2,2,1,1)
    58:      111010  (1,1,2,2)
    64:     1000000  (7)
   128:    10000000  (8)
   256:   100000000  (9)
   292:   100100100  (3,3,3)
   349:   101011101  (2,2,1,1,2,1)
   442:   110111010  (1,2,1,1,2,2)
   512:  1000000000  (10)
   586:  1001001010  (3,3,2,2)
   676:  1010100100  (2,2,3,3)
   697:  1010111001  (2,2,1,1,3,1)

The non-recursive version for partitions is A325755, counted by A325702.
These compositions are counted by A353391.
The version for partitions A353393, counted by A353426, w/o primes A353389.
The non-recursive version is A353402, counted by A353390.
The non-recursive consecutive case is A353432, counted by A353392.
The consecutive case is A353696, counted by A353430.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order, rev A228351, run-lens A333769.
A329738 counts uniform compositions, partitions A047966.
Statistics of standard compositions:
- Length is A000120, sum A070939.
- Runs are counted by A124767, distinct A351014.
- Subsequences are counted by A334299, contiguous A124770/A124771.
- Runs-resistance is A333628.
Classes of standard compositions:
- Partitions are A114994, multisets A225620, strict A333255, sets A333256.
- Constant compositions are A272919, counted by A000005.
- Golomb rulers are A333222, counted by A169942.
- Knapsack compositions are A333223, counted by A325676.
- Anti-runs are A333489, counted by A003242.
Cf. A032020, A044813, A114640, A165413, A181819, A329739, A318928, A325705, A333224, A353427, A353403.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
rorQ[y_]:=Length[y]<=1||MemberQ[Subsets[y],Length/@Split[y]]&& rorQ[Length/@Split[y]];
Select[Range[0,100],rorQ[stc[#]]&]

A353432 Numbers k such that the k-th composition in standard order has its own run-lengths as a consecutive subsequence.

Original entry on oeis.org

0, 1, 10, 21, 26, 43, 58, 107, 117, 174, 186, 292, 314, 346, 348, 349, 373, 430, 442, 570, 585, 586, 629, 676, 696, 697, 804, 826, 860, 861, 885, 1082, 1141, 1173, 1210, 1338, 1387, 1392, 1393, 1394, 1396, 1594, 1653, 1700, 1720, 1721, 1882, 2106, 2165, 2186

Offset: 1

Gus Wiseman, May 16 2022

nonn

First differs from A353402 (the non-consecutive version) in lacking 53.

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 initial terms, their binary expansions, and the corresponding standard compositions:
     0:          0  ()
     1:          1  (1)
    10:       1010  (2,2)
    21:      10101  (2,2,1)
    26:      11010  (1,2,2)
    43:     101011  (2,2,1,1)
    58:     111010  (1,1,2,2)
   107:    1101011  (1,2,2,1,1)
   117:    1110101  (1,1,2,2,1)
   174:   10101110  (2,2,1,1,2)
   186:   10111010  (2,1,1,2,2)
   292:  100100100  (3,3,3)
   314:  100111010  (3,1,1,2,2)
   346:  101011010  (2,2,1,2,2)
   348:  101011100  (2,2,1,1,3)
   349:  101011101  (2,2,1,1,2,1)
   373:  101110101  (2,1,1,2,2,1)
   430:  110101110  (1,2,2,1,1,2)
   442:  110111010  (1,2,1,1,2,2)

These compositions are counted by A353392.
This is the consecutive case of A353402, counted by A353390.
The non-consecutive recursive version is A353431, counted by A353391.
The recursive version is A353696, counted by A353430.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order, rev A228351, run-lens A333769.
A329738 counts uniform compositions, partitions A047966.
Statistics of standard compositions:
- Length is A000120, sum A070939.
- Runs are counted by A124767, distinct A351014.
- Subsequences are counted by A334299, contiguous A124770/A124771.
- Runs-resistance is A333628.
Classes of standard compositions:
- Partitions are A114994, strict A333255, rev A225620, strict rev A333256.
- Runs are A272919, counted by A000005.
- Golomb rulers are A333222, counted by A169942.
- Anti-runs are A333489, counted by A003242.
Cf. A044813, A165413, A181819, A318928, A325702, A325705, A325755, A333224, A333755, A353389, A353393, A353403.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
rorQ[y_]:=Length[y]==0||MemberQ[Join@@Table[Take[y,{i,j}],{i,Length[y]},{j,i,Length[y]}],Length/@Split[y]];
Select[Range[0,10000],rorQ[stc[#]]&]

A353929 Number of distinct sums of runs (of 0's or 1's) in the binary expansion of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 3, 2, 1, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 3, 2, 2, 3, 2, 1, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 3, 3, 3, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 2, 3, 3, 3, 2, 3, 2, 1, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 3, 3, 3, 3, 3, 2, 2, 2, 3, 2, 2, 3

Offset: 0

Gus Wiseman, Jun 26 2022

Assuming the binary digits are not all 1, this is one more than the number of different lengths of runs of 1's in the binary expansion of n.

			The binary expansion of 183 is (1,0,1,1,0,1,1,1), with runs (1), (0), (1,1), (0), (1,1,1), with sums 1, 0, 2, 0, 3, of which four are distinct, so a(183) = 4.

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

For lengths of all runs we have A165413, firsts A165933.
Numbers whose binary expansion has distinct runs are A175413.
For runs instead of run-sums we have A297770, firsts A350952.
For prime indices we have A353835, weak A353861, firsts A006939.
For standard compositions we have A353849, firsts A246534.
Positions of first appearances are A353930.
A005811 counts runs in binary expansion.
A044813 lists numbers with distinct run-lengths in binary expansion.
A318928 gives runs-resistance of binary expansion.
A351014 counts distinct runs in standard compositions.
Cf. A215203, A353743, A353832, A353847, A353932, A354579.

Table[Length[Union[Total/@Split[IntegerDigits[n,2]]]],{n,0,100}]

from itertools import groupby
def A353929(n): return len(set(sum(map(int,y[1])) for y in groupby(bin(n)[2:]))) # Chai Wah Wu, Jun 26 2022

A329750 Triangle read by rows where T(n,k) is the number of compositions of n >= 1 with runs-resistance n - k, 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 2, 6, 6, 1, 1, 0, 4, 9, 15, 3, 1, 0, 2, 16, 22, 22, 1, 1, 0, 0, 8, 37, 38, 41, 3, 1, 0, 0, 0, 26, 86, 69, 72, 2, 1, 0, 0, 0, 2, 78, 175, 124, 129, 3, 1, 0, 0, 0, 0, 14, 202, 367, 226, 213, 1, 1, 0, 0, 0, 0, 0, 52, 469, 750, 376, 395, 5, 1

Offset: 1

Gus Wiseman, Nov 21 2019

A composition of n is a finite sequence of positive integers with sum n.

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.

			Triangle begins:
   1
   1   1
   2   1   1
   2   3   2   1
   2   6   6   1   1
   0   4   9  15   3   1
   0   2  16  22  22   1   1
   0   0   8  37  38  41   3   1
   0   0   0  26  86  69  72   2   1
   0   0   0   2  78 175 124 129   3   1
   0   0   0   0  14 202 367 226 213   1   1
   0   0   0   0   0  52 469 750 376 395   5   1
Row n = 6 counts the following compositions:
  (1,1,3,1)    (1,1,4)      (1,5)      (3,3)          (6)
  (1,3,1,1)    (4,1,1)      (2,4)      (2,2,2)
  (1,1,1,2,1)  (1,1,1,3)    (4,2)      (1,1,1,1,1,1)
  (1,2,1,1,1)  (1,2,2,1)    (5,1)
               (2,1,1,2)    (1,2,3)
               (3,1,1,1)    (1,3,2)
               (1,1,1,1,2)  (1,4,1)
               (1,1,2,1,1)  (2,1,3)
               (2,1,1,1,1)  (2,3,1)
                            (3,1,2)
                            (3,2,1)
                            (1,1,2,2)
                            (1,2,1,2)
                            (2,1,2,1)
                            (2,2,1,1)

Row sums are A000079.
Column sums are A329768.
The version with rows reversed is A329744.
Cf. A000740, A008965, A098504, A242882, A318928, A329745, A329746, A329747, A329767.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==n-k&]],{n,10},{k,n}]

A329864 Number of compositions of n with the same runs-resistance as cuts-resistance.

Original entry on oeis.org

1, 0, 0, 0, 0, 2, 5, 10, 17, 27, 68, 107, 217, 420, 884, 1761, 3679, 7469, 15437, 31396, 64369

Offset: 0

Gus Wiseman, Nov 23 2019

A composition of n is a finite sequence of positive integers summing to n.
For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined to be the number of applications required to reach a singleton.
For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.

			The a(5) = 2 through a(8) = 17 compositions:
  (1112)  (1113)   (1114)    (1115)
  (2111)  (1122)   (1222)    (1133)
          (2211)   (2221)    (3311)
          (3111)   (4111)    (5111)
          (11211)  (11122)   (11222)
                   (11311)   (11411)
                   (21112)   (12221)
                   (22111)   (21113)
                   (111121)  (22211)
                   (121111)  (31112)
                             (111131)
                             (111221)
                             (112112)
                             (112211)
                             (122111)
                             (131111)
                             (211211)
For example, the runs-resistance of (111221) is 3 because we have: (111221) -> (321) -> (111) -> (3), while the cuts-resistance is also 3 because we have: (111221) -> (112) -> (1) -> (), so (111221) is counted under a(8).

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

The version for binary expansion is A329865.
Compositions counted by runs-resistance are A329744.
Compositions counted by cuts-resistance are A329861.
Compositions with runs-resistance = cuts-resistance minus 1 are A329869.
Cf. A003242, A098504, A114901, A242882, A318928, A319411, A319416, A319420, A319421, A329867, A329868.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==degdep[#]&]],{n,0,10}]

A329867 Runs-resistance minus cuts-resistance of the binary expansion of n.

Original entry on oeis.org

0, -1, 1, -1, 1, 1, 1, -2, 0, 1, 1, 2, 0, 2, 0, -3, -1, 0, 3, 2, 2, 1, 3, 1, 0, 2, 2, 0, 0, 1, -1, -4, -2, -1, 2, 0, 0, 3, 2, 0, 1, 3, 1, 2, 1, 2, 2, 0, -1, 0, 1, 0, 2, 2, 0, -1, -1, 0, 1, -1, -1, 0, -2, -5, -3, -2, 1, -1, -1, 2, 0, 1, -1, 0, 3, 4, 2, 3, 0

Offset: 0

Gus Wiseman, Nov 23 2019

sign

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined to be the number of applications required to reach a singleton.

For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.

			The sequence of binary expansions together with their runs-resistances and cuts-resistances, and their differences, begins:
   0      (): 0 - 0 =  0
   1     (1): 0 - 1 = -1
   2    (10): 2 - 1 =  1
   3    (11): 1 - 2 = -1
   4   (100): 3 - 2 =  1
   5   (101): 2 - 1 =  1
   6   (110): 3 - 2 =  1
   7   (111): 1 - 3 = -2
   8  (1000): 3 - 3 =  0
   9  (1001): 3 - 2 =  1
  10  (1010): 2 - 1 =  1
  11  (1011): 4 - 2 =  2
  12  (1100): 2 - 2 =  0
  13  (1101): 4 - 2 =  2
  14  (1110): 3 - 3 =  0
  15  (1111): 1 - 4 = -3
  16 (10000): 3 - 4 = -1
  17 (10001): 3 - 3 =  0
  18 (10010): 5 - 2 =  3
  19 (10011): 4 - 2 =  2
  20 (10100): 4 - 2 =  2

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Positions of 0's are A329865.
Positions of -1's are A329866.
Sorted positions of first appearances are A329868.
Compositions with runs-resistance equal to cuts-resistance are A329864.
Compositions with runs-resistance = cuts-resistance minus 1 are A329869.
Runs-resistance of binary expansion is A318928.
Cuts-resistance of binary expansion is A319416.
Compositions counted by runs-resistance are A329744.
Compositions counted by cuts-resistance are A329861.
Binary words counted by runs-resistance are A319411 and A329767.
Binary words counted by cuts-resistance are A319421 and A329860.
Cf. A000975, A003242, A107907, A164707, A329738.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Table[If[n==0,0,runsres[IntegerDigits[n,2]]-degdep[IntegerDigits[n,2]]],{n,0,100}]

For n > 1, a(2^n) = 3 - n.

For n > 1, a(2^n - 1) = 1 - n.

A353427 Numbers k such that the k-th composition in standard order has all run-lengths > 1.

Original entry on oeis.org

0, 3, 7, 10, 15, 31, 36, 42, 43, 58, 63, 87, 122, 127, 136, 147, 170, 171, 175, 228, 234, 235, 250, 255, 292, 295, 343, 351, 471, 484, 490, 491, 506, 511, 528, 547, 586, 591, 676, 682, 683, 687, 698, 703, 904, 915, 938, 939, 943, 983, 996, 1002, 1003, 1018

Offset: 1

Gus Wiseman, May 16 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 terms and corresponding compositions begin:
     0: ()
     3: (1,1)
     7: (1,1,1)
    10: (2,2)
    15: (1,1,1,1)
    31: (1,1,1,1,1)
    36: (3,3)
    42: (2,2,2)
    43: (2,2,1,1)
    58: (1,1,2,2)
    63: (1,1,1,1,1,1)
    87: (2,2,1,1,1)
   122: (1,1,1,2,2)
   127: (1,1,1,1,1,1,1)

The version for partitions is A001694, counted by A007690.
The version for parts instead of lengths is A022340, counted by A212804.
These compositions are counted by A114901.
A subset of A348612 (counted by A261983).
The case of all run-lengths = 2 is A351011.
The case of all run-lengths > 2 is counted by A353400.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order, reverse A228351.
Statistics of standard compositions:
- Length is A000120, sum A070939.
- Runs are counted by A124767.
- Runs-resistance is A333628.
- Run-lengths are A333769.
Cf. A044813, A128695, A165413, A240085, A244164, A274174, A318928, A333489, A333755, A353402, A353432.

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

A353854 Length of the trajectory of the composition run-sum transformation (condensation) of the n-th composition in standard order.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 3, 2, 3, 2, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 2, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 3, 1, 1, 3, 2, 1, 1, 1, 2, 2, 2, 2

Offset: 0

Gus Wiseman, Jun 01 2022

nonn

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4). The run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353847) until the rank of an anti-run is reached. For example, the trajectory 11 -> 10 -> 8 corresponds to the trajectory (2,1,1) -> (2,2) -> (4), with length a(11) = 3.

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 trajectory of 94685 and the a(94685) = 5 corresponding compositions:
  94685: (2,1,1,4,1,1,2,1,1,2,1)
  86357: (2,2,4,2,2,2,2,1)
  69889: (4,4,8,1)
  65793: (8,8,1)
  65537: (16,1)

Positions of first appearances are A072639.
Positions of 1's are A333489, counted by A003242 (complement A261983).
The version for partitions is A353841.
The last part of the same trajectory is A353855.
This is the rank statistic counted by A353859.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order.
A318928 gives runs-resistance of binary expansion.
A333627 represents the run-lengths of standard compositions.
A353832 represents the run-sum transformation of a partition.
A353840-A353846 pertain to the partition run-sum trajectory.
A353847 represents the run-sum transformation of a composition.
A353853-A353859 pertain to the composition run-sum trajectory.
A353932 lists run-sums of standard compositions, represented by A353847.
Cf. A237685, A238279, A304442, A325277, A333755, A353848, A353849, A353850, A353852, A353860.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[FixedPointList[Total/@Split[#]&,stc[n]]]-1,{n,0,100}]

cp's OEIS Frontend

A353402 Numbers k such that the k-th composition in standard order has its own run-lengths as a subsequence (not necessarily consecutive).

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A329862 Positive integers whose binary expansion has cuts-resistance 2.

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A353431 Numbers k such that the k-th composition in standard order is empty, a singleton, or has its own run-lengths as a subsequence (not necessarily consecutive) that is already counted.

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A353432 Numbers k such that the k-th composition in standard order has its own run-lengths as a consecutive subsequence.

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A353929 Number of distinct sums of runs (of 0's or 1's) in the binary expansion of n.

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A329750 Triangle read by rows where T(n,k) is the number of compositions of n >= 1 with runs-resistance n - k, 1 <= k <= n.

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A329864 Number of compositions of n with the same runs-resistance as cuts-resistance.

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A329867 Runs-resistance minus cuts-resistance of the binary expansion of n.

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A353427 Numbers k such that the k-th composition in standard order has all run-lengths > 1.