A318928 -id:A318928 - cp's OEIS frontend

A353855 Last term of the trajectory of the composition run-sum transformation (condensation) of the n-th composition in standard order.

0, 1, 2, 2, 4, 5, 6, 4, 8, 9, 8, 8, 12, 13, 8, 8, 16, 17, 18, 18, 20, 17, 22, 20, 24, 25, 24, 24, 20, 17, 18, 16, 32, 33, 34, 34, 32, 37, 38, 32, 40, 41, 32, 34, 44, 45, 32, 40, 48, 49, 50, 50, 52, 49, 54, 52, 40, 41, 40, 32, 32, 37, 34, 32, 64, 65, 66, 66, 68

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, corresponding to (2,1,1) -> (2,2) -> (4), has last term a(11) = 8.

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 139 -> 138 -> 136 -> 128 ends with a(139) = 128.

The version for partitions is A353842.
This trajectory has length A353854, firsts A072639, partitions A353841.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order.
A318928 gives runs-resistance of binary expansion.
A325268 counts partitions by omicron, rank statistic A304465.
A333627 ranks the run-lengths of standard compositions.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353840-A353846 pertain to a partition's run-sum trajectory.
A353847 represents a composition's run-sums, partitions A353832.
A353853-A353859 pertain to a composition's run-sum trajectory.
Cf. A237685, A304442, A333381, A333489, A333755, A353848, A353849, A353850, A353852, A353860.

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

A353858 Number of integer compositions of n with run-sum trajectory ending in a singleton.

Original entry on oeis.org

0, 1, 2, 2, 5, 2, 8, 2, 20, 5, 8, 2, 78, 2, 8, 8, 223, 2, 179, 2, 142, 8, 8, 2, 4808

Offset: 0

Gus Wiseman, Jun 17 2022

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-sums (cf. A353847) until an anti-run composition (A003242) is reached. For example, the composition (2,2,1,1,2) is counted under a(8) because it has the following run-sum trajectory: (2,2,1,1,2) -> (4,2,2) -> (4,4) -> (8).

			The a(0) = 0 through a(8) = 20 compositions:
  .  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
          (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                       (112)            (222)                (224)
                       (211)            (1113)               (422)
                       (1111)           (2112)               (1124)
                                        (3111)               (2114)
                                        (11211)              (2222)
                                        (111111)             (4112)
                                                             (4211)
                                                             (11114)
                                                             (21122)
                                                             (22112)
                                                             (41111)
                                                             (111122)
                                                             (112112)
                                                             (211211)
                                                             (221111)
                                                             (1111211)
                                                             (1121111)
                                                             (11111111)

The version for partitions is A353845, ranked by A353844.
The trajectory itself is A353853, last part A353855.
The lengths of trajectories of standard compositions are A353854.
This is column k = 1 of A353856, for partitions A353843.
These compositions are ranked by A353857.
A011782 counts compositions.
A066099 lists compositions in standard order.
A238279 and A333755 count compositions by number of runs.
A275870 counts collapsible partitions, ranked by A300273.
A333489 ranks anti-runs, counted by A003242 (complement A261983).
A353840-A353846 pertain to partition run-sum trajectory.
A353847 represents the run-sums of a composition, partitions A353832.
A353851 counts compositions with equal run-sums, ranked by A353848.
A353859 counts compositions by length of run-sum trajectory.
A353860 counts collapsible compositions.
A353932 lists run-sums of standard compositions.
Cf. A072639, A237685, A304442, A304465, A318928, A353833, A353841, A353849, A353850, A353852.

Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n], Length[FixedPoint[Total/@Split[#]&,#]]==1&]],{n,0,15}]

A353930 Smallest number whose binary expansion has n distinct run-sums.

Original entry on oeis.org

1, 2, 11, 183, 5871, 375775, 48099263, 12313411455, 6304466665215, 6455773865180671, 13221424875890015231, 54154956291645502388223, 443637401941159955564326911, 7268555193403964711965932118015, 238176016577461115681699663643131903, 15609103422420491677315869156516292427775

Offset: 1

Gus Wiseman, Jun 07 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 terms, binary expansions, and standard compositions begin:
       1:                    1  (1)
       2:                   10  (2)
      11:                 1011  (2,1,1)
     183:             10110111  (2,1,2,1,1,1)
    5871:        1011011101111  (2,1,2,1,1,2,1,1,1,1)
  375775:  1011011101111011111  (2,1,2,1,1,2,1,1,1,2,1,1,1,1,1)

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

Essentially the same as A215203.
For prime indices instead of binary expansion we have A006939.
For lengths instead of sums of runs we have A165933 = firsts in A165413.
Numbers whose binary expansion has all distinct runs are A175413.
For standard compositions we have A246534, firsts of A353849.
For runs instead of run-sums we have A350952, firsts of A297770.
These are the positions of first appearances in A353929.
A005811 counts runs in binary expansion.
A242882 counts compositions with distinct multiplicities.
A318928 gives runs-resistance of binary expansion.
A351014 counts distinct runs in standard compositions.
A353835 counts partitions with all distinct run-sums, weak A353861.
A353864 counts rucksack partitions.
Cf. A044813, A073093, A181819, A304442, A353743, A353840, A353841, A353842, A353847, A353848, A353850, A353853, A353932, A354582.

qe=Table[Length[Union[Total/@Split[IntegerDigits[n,2]]]],{n,1,10000}];
Table[Position[qe,i][[1,1]],{i,Max@@qe}]

a(n) = {my(t=1); if(n==2, t<<=1, for(k=3, n, t = (t<Andrew Howroyd, Jan 01 2023

Offset corrected and terms a(7) and beyond from Andrew Howroyd, Jan 01 2023

A333629 Least k such that the runs-resistance of the k-th composition in standard order is n.

Original entry on oeis.org

1, 3, 5, 11, 27, 93, 859, 13789, 1530805, 1567323995

Offset: 0

Gus Wiseman, Mar 31 2020

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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.

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.

			The sequence together with the corresponding compositions begins:
        1: (1)
        3: (1,1)
        5: (2,1)
       11: (2,1,1)
       27: (1,2,1,1)
       93: (2,1,1,2,1)
      859: (1,2,2,1,2,1,1)
    13789: (1,2,2,1,1,2,1,1,2,1)
  1530805: (2,1,1,2,2,1,2,1,1,2,1,2,2,1)
For example, starting with 13789 and repeatedly applying A333627 gives: 13789 -> 859 -> 110 -> 29 -> 11 -> 6 -> 3 -> 2, corresponding to the compositions: (1,2,2,1,1,2,1,1,2,1) -> (1,2,2,1,2,1,1) -> (1,2,1,1,2) -> (1,1,2,1) -> (2,1,1) -> (1,2) -> (1,1) -> (2).

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

Positions of first appearances in A333628 = number of times applying A333627 to reach a power of 2, starting with n.
A subsequence of A333630.
All of the following pertain to compositions in standard order (A066099):
- The length is A000120.
- The partial sums from the right are A048793.
- The sum is A070939.
- Adjacent equal pairs are counted by A124762.
- Equal runs are counted by A124767.
- Strict compositions are ranked by A233564.
- The partial sums from the left are A272020.
- Constant compositions are ranked by A272919.
- Normal compositions are ranked by A333217.
- Heinz number is A333219.
- Anti-runs are counted by A333381.
- Adjacent unequal pairs are counted by A333382.
Cf. A029931, A098504, A114994, A225620, A228351, A238279, A242882, A318928, A329744, A329747, A333489.

nn=1000;
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
stcrun[n_]:=Total[2^(Accumulate[Reverse[Length/@Split[stc[n]]]])]/2;
seq=Table[Length[NestWhileList[stcrun,n,Length[stc[#]]>1&]]-1,{n,nn}];
Table[Position[seq,i][[1,1]],{i,Union[seq]}]

a(9) from Amiram Eldar, Aug 04 2025

A353428 Number of integer compositions of n with all parts and all run-lengths > 2.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 1, 0, 2, 0, 2, 4, 0, 0, 8, 3, 0, 10, 4, 4, 15, 4, 8, 24, 7, 8, 42, 16, 10, 59, 31, 27, 87, 37, 52, 149, 62, 66, 233, 121, 111, 342, 207, 204, 531, 308, 351, 864, 487, 536, 1373, 864, 865, 2057, 1440, 1509, 3232

Offset: 0

Gus Wiseman, May 16 2022

nonn

			The a(n) compositions for selected n:
  n=16:   n=18:     n=20:    n=21:      n=24:
----------------------------------------------------
  (4444)  (666)     (5555)   (777)      (888)
          (333333)  (44444)  (333444)   (6666)
                             (444333)   (333555)
                             (3333333)  (444444)
                                        (555333)
                                        (3333444)
                                        (4443333)
                                        (33333333)

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Allowing any multiplicities gives A078012, partitions A008483.
The version for no (instead of all) parts or run-lengths > 2 is A137200.
Allowing any parts gives A353400, partitions A100405.
The version for partitions is A353501, ranked by A353502.
The version for > 1 instead of > 2 is A353508, partitions A339222.
A003242 counts anti-run compositions, ranked by A333489.
A008466 counts compositions with some part > 2.
A011782 counts compositions.
A114901 counts compositions with no runs of length 1, ranked by A353427.
A128695 counts compositions with no run-lengths > 2.
A261983 counts non-anti-run compositions.
A335464 counts compositions with a run-length > 2.
Cf. A032020, A044813, A098859, A165413, A318928, A329738, A329739, A333755, A353390, A353429.

b:= proc(n, h) option remember; `if`(n=0, 1, add(
     `if`(i=h, 0, add(b(n-i*j, i), j=3..n/i)), i=3..n/3))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, May 18 2022

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[#,1|2]&&!MemberQ[Length/@Split[#],1|2]&]],{n,0,15}]

a(26)-a(66) from Alois P. Heinz, May 17 2022

A165933 Least integer, k, whose value is n in A165413.

Original entry on oeis.org

1, 4, 35, 536, 16775, 1060976, 135007759, 34460631520, 17617985239071, 18027600169142208, 36907002795598798911, 151143401509104346210176, 1238053384151947477501575295, 20283338091738780737237428502272, 664629209970464486086782992577855743

Offset: 1

Robert G. Wilson v, Sep 30 2009

An alternative name: The smallest number whose binary expansion has exactly n distinct run-lengths. - Gus Wiseman, Feb 21 2022

Term a(n) has one 1, followed by n 0's, then two 1's, (n-1) 0's, ..., up to n runs; see Python program. - Michael S. Branicky, Feb 22 2022

			a(1) in binary is 1, a(2) in binary is 100, a(3) in binary is 100011, a(4) in binary is 1000011000, etc.
From _Gus Wiseman_, Feb 21 2022: (Start)
The terms and their binary expansions begin:
  n              a(n)
  1:               1 =                                             1
  2:               4 =                                           100
  3:              35 =                                        100011
  4:             536 =                                    1000011000
  5:           16775 =                               100000110000111
  6:         1060976 =                         100000011000001110000
  7:       135007759 =                  1000000011000000111000001111
  8:     34460631520 =          100000000110000000111000000111100000
  9:  17617985239071 = 100000000011000000001110000000111100000011111
(End)

Michael S. Branicky, Table of n, a(n) for n = 1..81

A subset of A044813 (distinct run-lengths) and of A175413 (distinct runs).
These are the positions of first appearances in A165413.
The version for runs instead of run-lengths is A350952, firsts of A297770.
A000120 counts binary weight.
A005811 counts runs in binary expansion.
A242882 counts compositions with distinct multiplicities.
A318928 gives runs-resistance of binary expansion.
A351014 counts distinct runs in standard compositions.
Counting words with all distinct run-lengths:
- A032020 = binary expansions, for runs A351018.
- A329739 = compositions, for runs A351013.
- A351017 = binary words, for runs A351016.
- A351292 = patterns, for runs A351200.
Cf. A003242, A070939, A085207, A098859, A325545, A328592, A351015, A351202, A351203, A351291.

g[n_] := Table[ {Table[1, {i}], Table[0, {n - i + 1}]}, {i, Floor[(n + If[ OddQ@n, 1, 0])/2]}]; f[n_] := FromDigits[ If[ OddQ@n, Flatten@ Most@ Flatten[ g@n, 1], Flatten@ g@n], 2]; Array[f, 14]
s=Table[Length[Union[Length/@Split[IntegerDigits[n,2]]]],{n,0,1000}]; Table[Position[s,k][[1,1]]-1,{k,Union[s]}] (* Gus Wiseman, Feb 21 2022 *)

def a(n): # returns term by construction
    if n == 1: return 1
    q, r = divmod(n+1, 2)
    s = "".join("1"*i + "0"*(n+1-i) for i in range(1, q+1))
    if r == 0: s = s.rstrip("0")
    return int(s, 2)
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Feb 22 2022

a(15) and beyond from Michael S. Branicky, Feb 22 2022

A329866 Numbers whose binary expansion has its runs-resistance equal to its cuts-resistance minus 1.

Original entry on oeis.org

1, 3, 16, 30, 33, 48, 55, 56, 59, 60, 67, 68, 72, 79, 95, 97, 110, 112, 118, 120, 121, 125, 134, 135, 137, 143, 145, 158, 160, 195, 196, 219, 220, 225, 231, 241, 250, 258, 270, 280, 286, 291, 292, 315, 316, 351, 381, 382, 390, 391, 393, 399, 415, 416, 431, 432

Offset: 1

Gus Wiseman, Nov 23 2019

nonn

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 terms together with their binary expansions begins:
    1:         1
    3:        11
   16:     10000
   30:     11110
   33:    100001
   48:    110000
   55:    110111
   56:    111000
   59:    111011
   60:    111100
   67:   1000011
   68:   1000100
   72:   1001000
   79:   1001111
   95:   1011111
   97:   1100001
  110:   1101110
  112:   1110000
  118:   1110110
  120:   1111000
For example, 79 has runs-resistance 3 because we have (1001111) -> (124) -> (111) -> (3), while the cuts-resistance is 4 because we have (1001111) -> (0111) -> (11) -> (1) -> (), so 79 is in the sequence.

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

Positions of -1's in A329867.
The version for runs-resistance equal to cuts-resistance is A329865.
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, A329868.

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

A329868 Sorted positions of first appearances in A329867 (difference between the runs-resistance and the cuts-resistance of binary expansion) of each element in the image.

Original entry on oeis.org

0, 1, 2, 7, 11, 15, 18, 31, 63, 75, 127, 255, 511, 1023, 1234, 2047, 4095, 8191, 9638, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607

Offset: 1

Gus Wiseman, Nov 23 2019

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 terms together with their binary expansions begins:
      0:
      1:                1
      2:               10
      7:              111
     11:             1011
     15:             1111
     18:            10010
     31:            11111
     63:           111111
     75:          1001011
    127:          1111111
    255:         11111111
    511:        111111111
   1023:       1111111111
   1234:      10011010010
   2047:      11111111111
   4095:     111111111111
   8191:    1111111111111
   9638:   10010110100110
  16383:   11111111111111
  32767:  111111111111111
  65535: 1111111111111111

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

Sorted positions of first appearances in A329867.
Compositions with runs-resistance equal to cuts-resistance are A329864.
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, A329865, A329866.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
das=Table[If[n==0,0,runsres[IntegerDigits[n,2]]-degdep[IntegerDigits[n,2]]],{n,0,1000000}];
Table[Position[das,i][[1,1]]-1,{i,First/@Gather[das]}]

A333630 Least STC-number of a composition whose sequence of run-lengths has STC-number n.

Original entry on oeis.org

0, 1, 3, 5, 7, 14, 11, 13, 15, 30, 43, 29, 23, 46, 27, 45, 31, 62, 122, 61, 87, 117, 59, 118, 47, 94, 107, 93, 55, 110, 91, 109, 63, 126, 250, 125, 343, 245, 123, 246, 175, 350, 235, 349, 119, 238, 347, 237, 95, 190, 378, 189, 215, 373, 187, 374, 111, 222, 363

Offset: 0

Gus Wiseman, Mar 31 2020

nonn

All terms belong to A003754.

A composition of n is a finite sequence of positive integers summing to n. The composition with STC-number k (row k of 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: ()
   1: (1)
   3: (1,1)
   5: (2,1)
   7: (1,1,1)
  14: (1,1,2)
  11: (2,1,1)
  13: (1,2,1)
  15: (1,1,1,1)
  30: (1,1,1,2)
  43: (2,2,1,1)
  29: (1,1,2,1)
  23: (2,1,1,1)
  46: (2,1,1,2)
  27: (1,2,1,1)
  45: (2,1,2,1)
  31: (1,1,1,1,1)
  62: (1,1,1,1,2)

Position of first appearance of n in A333627.
All of the following pertain to compositions in standard order (A066099):
- The length is A000120.
- Compositions without terms > 2 are A003754.
- Compositions without ones are ranked by A022340.
- The partial sums from the right are A048793.
- The sum is A070939.
- Adjacent equal pairs are counted by A124762.
- Equal runs are counted by A124767.
- Strict compositions are ranked by A233564.
- The partial sums from the left are A272020.
- Constant compositions are ranked by A272919.
- Normal compositions are ranked by A333217.
- Heinz number is A333219.
- Anti-runs are counted by A333381.
- Adjacent unequal pairs are counted by A333382.
- Runs-resistance is A333628.
- First appearances of run-resistances are A333629.
Cf. A029931, A098504, A114994, A225620, A228351, A238279, A242882, A318928, A329744, A329747, A333489.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
seq=Table[Total[2^(Accumulate[Reverse[Length/@Split[stc[n]]]])]/2,{n,0,1000}];
Table[Position[seq,i][[1,1]],{i,First[Split[Union[seq],#1+1==#2&]]}]-1

A337565 Irregular triangle read by rows where row k is the sequence of maximal anti-run lengths in the k-th composition in standard order.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 3, 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 3, 3, 2, 1, 1, 2, 3, 1, 1, 1, 1, 2, 1, 3, 4, 2, 2, 2, 1, 1, 1, 2, 3, 3

Offset: 0

Gus Wiseman, Sep 07 2020

An anti-run is a sequence with no adjacent equal parts.

A composition of n is a finite sequence of positive integers summing to n. 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 first column below lists various selected n; the second column gives the corresponding composition; the third column gives the corresponding row of the triangle, i.e., the anti-run lengths.
    1:           (1) -> (1)
    3:         (1,1) -> (1,1)
    5:         (2,1) -> (2)
    7:       (1,1,1) -> (1,1,1)
   11:       (2,1,1) -> (2,1)
   13:       (1,2,1) -> (3)
   14:       (1,1,2) -> (1,2)
   15:     (1,1,1,1) -> (1,1,1,1)
   23:     (2,1,1,1) -> (2,1,1)
   27:     (1,2,1,1) -> (3,1)
   29:     (1,1,2,1) -> (1,3)
   30:     (1,1,1,2) -> (1,1,2)
   31:   (1,1,1,1,1) -> (1,1,1,1,1)
   43:     (2,2,1,1) -> (1,2,1)
   45:     (2,1,2,1) -> (4)
   46:     (2,1,1,2) -> (2,2)
   47:   (2,1,1,1,1) -> (2,1,1,1)
   55:   (1,2,1,1,1) -> (3,1,1)
   59:   (1,1,2,1,1) -> (1,3,1)
   61:   (1,1,1,2,1) -> (1,1,3)
   62:   (1,1,1,1,2) -> (1,1,1,2)
   63: (1,1,1,1,1,1) -> (1,1,1,1,1,1)
For example, the 119th composition is (1,1,2,1,1,1), with maximal anti-runs ((1),(1,2,1),(1),(1)), so row 119 is (1,3,1,1).

A000120 gives row sums.
A333381 gives row lengths.
A333769 is the version for runs.
A003242 counts anti-run compositions.
A011782 counts compositions.
A106351 counts anti-run compositions by length.
A329744 is a triangle counting compositions by runs-resistance.
A333755 counts compositions by number of runs.
All of the following pertain to compositions in standard order (A066099):
- Sum is A070939.
- Adjacent equal pairs are counted by A124762.
- Runs are counted by A124767.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Patterns are A333217.
- Heinz number is A333219.
- Anti-runs are counted by A333381.
- Anti-run compositions are A333489.
- Runs-resistance is A333628.
- Combinatory separations are A334030.
Cf. A106356, A113835, A114994, A124767, A181819, A228351, A238279, A318928, A333216, A333627, A333630.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length/@Split[stc[n],UnsameQ],{n,0,50}]

cp's OEIS Frontend

A353855 Last term of the trajectory of the composition run-sum transformation (condensation) of the n-th composition in standard order.

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A353858 Number of integer compositions of n with run-sum trajectory ending in a singleton.

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A353930 Smallest number whose binary expansion has n distinct run-sums.

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A333629 Least k such that the runs-resistance of the k-th composition in standard order is n.

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A353428 Number of integer compositions of n with all parts and all run-lengths > 2.

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A165933 Least integer, k, whose value is n in A165413.

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A329866 Numbers whose binary expansion has its runs-resistance equal to its cuts-resistance minus 1.

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A329868 Sorted positions of first appearances in A329867 (difference between the runs-resistance and the cuts-resistance of binary expansion) of each element in the image.

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