A124767 -id:A124767 - cp's OEIS frontend

A333627 The a(n)-th composition in standard order is the sequence of run-lengths of the n-th composition in standard order.

0, 1, 1, 2, 1, 3, 3, 4, 1, 3, 2, 6, 3, 7, 5, 8, 1, 3, 3, 6, 3, 5, 7, 12, 3, 7, 6, 14, 5, 11, 9, 16, 1, 3, 3, 6, 2, 7, 7, 12, 3, 7, 4, 10, 7, 15, 13, 24, 3, 7, 7, 14, 7, 13, 15, 28, 5, 11, 10, 22, 9, 19, 17, 32, 1, 3, 3, 6, 3, 7, 7, 12, 3, 5, 6, 14, 7, 15, 13

Offset: 0

Gus Wiseman, Mar 30 2020

nonn

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. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The standard compositions and their run-lengths:
       0 ~ () -> () ~ 0
      1 ~ (1) -> (1) ~ 1
      2 ~ (2) -> (1) ~ 1
     3 ~ (11) -> (2) ~ 2
      4 ~ (3) -> (1) ~ 1
     5 ~ (21) -> (11) ~ 3
     6 ~ (12) -> (11) ~ 3
    7 ~ (111) -> (3) ~ 4
      8 ~ (4) -> (1) ~ 1
     9 ~ (31) -> (11) ~ 3
    10 ~ (22) -> (2) ~ 2
   11 ~ (211) -> (12) ~ 6
    12 ~ (13) -> (11) ~ 3
   13 ~ (121) -> (111) ~ 7
   14 ~ (112) -> (21) ~ 5
  15 ~ (1111) -> (4) ~ 8
     16 ~ (5) -> (1) ~ 1
    17 ~ (41) -> (11) ~ 3
    18 ~ (32) -> (11) ~ 3
   19 ~ (311) -> (12) ~ 6

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

Positions of first appearances are 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.
- 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;
Table[Total[2^(Accumulate[Reverse[Length/@Split[stc[n]]]])]/2,{n,0,30}]

A000120(n) = A070939(a(n)).

A000120(a(n)) = A124767(n).

A124765 Number of monotonically decreasing runs for compositions in standard order.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 3

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

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. a(n) is the number of maximal weakly decreasing runs in this composition. Alternatively, a(n) is one plus the number of strict ascents in the same composition. For example, the weakly decreasing runs of the 1234567th composition are ((3,2,1),(2,2,1),(2),(5,1,1,1)), so a(1234567) = 4. The 3 strict ascents together with the weak descents are: 3 >= 2 >= 1 < 2 >= 2 >= 1 < 2 < 5 >= 1 >= 1 >= 1. - Gus Wiseman, Apr 08 2020

			Composition number 11 is 2,1,1; the decreasing runs are 2,1,1; so a(11) = 1.
The table starts:
  0
  1
  1 1
  1 1 2 1
  1 1 1 1 2 2 2 1
  1 1 1 1 2 1 2 1 2 2 2 2 2 2 2 1
  1 1 1 1 1 1 2 1 2 2 1 1 2 2 2 1 2 2 2 2 3 2 3 2 2 2 2 2 2 2 2 1

Antti Karttunen, Table of n, a(n) for n = 0..16383

Cf. A066099, A124760, A011782 (row lengths).
Compositions of n with k strict ascents are A238343.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Weakly decreasing compositions are A114994.
- Weakly decreasing runs are counted by A124765.
- Weakly increasing runs are counted by A124766.
- Equal runs are counted by A124767.
- Strictly increasing runs are counted by A124768.
- Strictly decreasing runs are counted by A124769.
- Weakly increasing compositions are A225620.
- Reverse is A228351 (triangle).
- Strict compositions are A233564.
- Constant compositions are A272919.
- Normal compositions are A333217.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
- Anti-runs are counted by A333381.
- Adjacent unequal pairs are counted by A333382.
- Anti-runs are A333489.
- Runs-resistance is A333628.
Cf. A124761, A124763, A124764, A233249, A333213, A333380.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Split[stc[n],GreaterEqual]],{n,0,100}] (* Gus Wiseman, Apr 08 2020 *)

a(0) = 0, a(n) = A124760(n) + 1 for n > 0.

A335465 Number of minimal normal patterns avoided by the n-th composition in standard order (A066099).

Original entry on oeis.org

1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 12, 4, 3, 3, 3, 3, 4, 3, 4, 12, 4, 3, 12, 4, 12, 4, 12, 4, 3, 3, 3, 3, 4, 3, 3, 6, 4, 3, 6, 3, 3, 6, 10, 10, 4, 3, 12, 6, 12, 3, 10, 10, 12, 4, 12, 3, 12, 4, 12, 4, 3, 3, 3, 3, 4, 3, 3, 6

Offset: 0

Gus Wiseman, Jun 20 2020

These patterns comprise the basis of the class of patterns generated by this composition.
We define a (normal) pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).
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 bases of classes generated by (), (1), (2,1,1), (3,1,2), (2,1,2,1), and (1,2,1), corresponding to n = 0, 1, 11, 38, 45, 13, are the respective columns below.
  (1)  (1,1)  (1,2)    (1,1)    (1,1,1)    (1,1,1)
       (1,2)  (1,1,1)  (1,2,3)  (1,1,2)    (1,1,2)
       (2,1)  (2,2,1)  (1,3,2)  (1,2,2)    (1,2,2)
              (3,2,1)  (2,1,3)  (1,2,3)    (1,2,3)
                       (2,3,1)  (1,3,2)    (1,3,2)
                       (3,2,1)  (2,1,3)    (2,1,1)
                                (2,3,1)    (2,1,2)
                                (3,1,2)    (2,1,3)
                                (3,2,1)    (2,2,1)
                                (2,2,1,1)  (2,3,1)
                                           (3,1,2)
                                           (3,2,1)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

Patterns matched by standard compositions are counted by A335454.
Patterns matched by compositions of n are counted by A335456(n).
The version for Heinz numbers of partitions is A335550.
Patterns are counted by A000670 and ranked by A333217.
Knapsack compositions are counted by A325676 and ranked by A333223.
The n-th composition has A334299(n) distinct subsequences.
Cf. A056986, A124767, A124770, A124771, A269134, A333218, A333257, A335549.

A373948 Run-compression encoded as a transformation of compositions in standard order.

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 6, 1, 8, 9, 2, 5, 12, 13, 6, 1, 16, 17, 18, 9, 20, 5, 22, 5, 24, 25, 6, 13, 12, 13, 6, 1, 32, 33, 34, 17, 4, 37, 38, 9, 40, 41, 2, 5, 44, 45, 22, 5, 48, 49, 50, 25, 52, 13, 54, 13, 24, 25, 6, 13, 12, 13, 6, 1, 64, 65, 66, 33, 68, 69, 70, 17, 72

Offset: 0

Gus Wiseman, Jun 24 2024

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.
We define the (run-) compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has compression (1,2,1).
For the present sequence, the a(n)-th composition in standard order is obtained by compressing the n-th composition in standard order.

			The standard compositions and their compressions begin:
   0: ()        -->  0: ()
   1: (1)       -->  1: (1)
   2: (2)       -->  2: (2)
   3: (1,1)     -->  1: (1)
   4: (3)       -->  4: (3)
   5: (2,1)     -->  5: (2,1)
   6: (1,2)     -->  6: (1,2)
   7: (1,1,1)   -->  1: (1)
   8: (4)       -->  8: (4)
   9: (3,1)     -->  9: (3,1)
  10: (2,2)     -->  2: (2)
  11: (2,1,1)   -->  5: (2,1)
  12: (1,3)     --> 12: (1,3)
  13: (1,2,1)   --> 13: (1,2,1)
  14: (1,1,2)   -->  6: (1,2)
  15: (1,1,1,1) -->  1: (1)

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

Positions of 1's are A000225.
The image is A333489, counted by A003242.
Sum of standard composition for a(n) is given by A373953, length A124767.
A037201 gives compression of first differences of primes, halved A373947.
A066099 lists the parts of all compositions in standard order.
A114901 counts compositions with no isolated parts.
A116861 counts partitions by compressed sum, by length A116608.
A240085 counts compositions with no unique parts.
A333755 counts compositions by compressed length.
A373949 counts compositions by compressed sum, opposite A373951.
Cf. A106356, A124762, A238130, A238279, A272919, A333381, A333382, A333627, A373952, A373954.

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]]],{n,0,30}]

A029837(a(n)) = A373953(n).

A000120(a(n)) = A124767(n).

A124769 Number of strictly decreasing runs for compositions in standard order.

Original entry on oeis.org

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

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

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. a(n) is the number of maximal strictly decreasing runs in this composition. Alternatively, a(n) is one plus the number of weak ascents in the same composition. For example, the strictly decreasing runs of the 1234567th composition are ((3,2,1),(2),(2,1),(2),(5,1),(1),(1)), so a(1234567) = 7. The 6 weak ascents together with the strict descents are: 3 > 2 > 1 <= 2 <= 2 > 1 <= 2 <= 5 > 1 <= 1 <= 1. - Gus Wiseman, Apr 08 2020

			Composition number 11 is 2,1,1; the strictly increasing runs are 2,1; 1; so a(11) = 2.
The table starts:
  0
  1
  1 2
  1 1 2 3
  1 1 2 2 2 2 3 4
  1 1 1 2 2 2 2 3 2 2 3 3 3 3 4 5
  1 1 1 2 2 1 2 3 2 2 3 3 2 2 3 4 2 2 2 3 3 3 3 4 3 3 4 4 4 4 5 6

Cf. A066099, A124764, A011782 (row lengths).
Compositions of n with k weak ascents are A333213.
Positions of ones are A333256.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Partial sums from the right are A048793 (triangle).
- Sum is A070939.
- Weakly decreasing compositions are A114994.
- Weakly decreasing runs are counted by A124765.
- Weakly increasing runs are counted by A124766.
- Equal runs are counted by A124767.
- Strictly increasing runs are counted by A124768.
- Strictly decreasing runs are counted by A124769 (this sequence).
- Reversed initial intervals A164894.
- Weakly increasing compositions are A225620.
- Reverse is A228351 (triangle).
- Strict compositions are A233564.
- Constant compositions are A272919.
- Normal compositions are A333217.
- Permutations are A333218.
- Heinz number is A333219.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
- Anti-runs are A333489.
Cf. A124760, A124761, A124763, A233249, A238343.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Split[stc[n],Greater]],{n,0,100}] (* Gus Wiseman, Apr 08 2020 *)

a(0) = 0, a(n) = A124764(n) + 1 for n > 0.

A344618 Reverse-alternating sums of standard compositions (A066099). Alternating sums of the compositions ranked by A228351.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jun 03 2021

Up to sign, same as A124754.
The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
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 of nonnegative integers together with the corresponding standard compositions and their reverse-alternating sums begins:
  0:     () ->  0    15: (1111) ->  0    30:  (1112) ->  1
  1:    (1) ->  1    16:    (5) ->  5    31: (11111) ->  1
  2:    (2) ->  2    17:   (41) -> -3    32:     (6) ->  6
  3:   (11) ->  0    18:   (32) -> -1    33:    (51) -> -4
  4:    (3) ->  3    19:  (311) ->  3    34:    (42) -> -2
  5:   (21) -> -1    20:   (23) ->  1    35:   (411) ->  4
  6:   (12) ->  1    21:  (221) ->  1    36:    (33) ->  0
  7:  (111) ->  1    22:  (212) ->  3    37:   (321) ->  2
  8:    (4) ->  4    23: (2111) -> -1    38:   (312) ->  4
  9:   (31) -> -2    24:   (14) ->  3    39:  (3111) -> -2
  10:  (22) ->  0    25:  (131) -> -1    40:    (24) ->  2
  11: (211) ->  2    26:  (122) ->  1    41:   (231) ->  0
  12:  (13) ->  2    27: (1211) ->  1    42:   (222) ->  2
  13: (121) ->  0    28:  (113) ->  3    43:  (2211) ->  0
  14: (112) ->  2    29: (1121) -> -1    44:   (213) ->  4
Triangle begins (row lengths A011782):
  0
  1
  2  0
  3 -1  1  1
  4 -2  0  2  2  0  2  0
  5 -3 -1  3  1  1  3 -1  3 -1  1  1  3 -1  1  1

Up to sign, same as the reverse version A124754.
The version for Heinz numbers of partitions is A344616.
Positions of zeros are A344619.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A116406 counts compositions with alternating sum >= 0.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
All of the following pertain to compositions in standard order:
- The length is A000120.
- Converting to reversed ranking gives A059893.
- The rows are A066099.
- The sum is A070939.
- The runs are counted by A124767.
- The reversed version is A228351.
- Strict compositions are ranked by A233564.
- Constant compositions are ranked by A272919.
- The Heinz number is A333219.
- Anti-run compositions are ranked by A333489.
Cf. A000070, A000097, A003242, A028260, A119899, A239830, A344605, A344607, A344608, A344650, A344739.

sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]]
Table[sats[stc[n]],{n,0,100}]

A335454 Number of normal patterns matched by the n-th composition in standard order (A066099).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 5, 3, 6, 5, 5, 2, 3, 3, 5, 3, 5, 6, 7, 3, 6, 5, 9, 5, 9, 7, 6, 2, 3, 3, 5, 3, 4, 5, 7, 3, 5, 4, 7, 5, 10, 9, 9, 3, 6, 5, 9, 4, 9, 10, 12, 5, 9, 7, 13, 7, 12, 9, 7, 2, 3, 3, 5, 3, 4, 5, 7, 3, 5, 5, 7, 6, 10, 9, 9, 3, 5, 6, 8, 5

Offset: 0

Gus Wiseman, Jun 14 2020

We define a (normal) pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

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 a(n) patterns for n = 0, 1, 3, 7, 11, 13, 23, 83, 27, 45:
  0:  1:   11:   111:   211:   121:   2111:   2311:   1211:   2121:
---------------------------------------------------------------------
  ()  ()   ()    ()     ()     ()     ()      ()      ()      ()
      (1)  (1)   (1)    (1)    (1)    (1)     (1)     (1)     (1)
           (11)  (11)   (11)   (11)   (11)    (11)    (11)    (11)
                 (111)  (21)   (12)   (21)    (12)    (12)    (12)
                        (211)  (21)   (111)   (21)    (21)    (21)
                               (121)  (211)   (211)   (111)   (121)
                                      (2111)  (231)   (121)   (211)
                                              (2311)  (211)   (212)
                                                      (1211)  (221)
                                                              (2121)

John Tyler Rascoe, Table of n, a(n) for n = 0..8192
Wikipedia, Permutation pattern
Gus Wiseman, Statistics, classes, and transformations of standard compositions
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

References found in the links are not all included here.
Summing over indices with binary length n gives A335456(n).
The contiguous case is A335458.
The version for Heinz numbers of partitions is A335549.
Patterns are counted by A000670 and ranked by A333217.
The n-th composition has A124771(n) distinct consecutive subsequences.
Knapsack compositions are counted by A325676 and ranked by A333223.
The n-th composition has A333257(n) distinct subsequence-sums.
The n-th composition has A334299(n) distinct subsequences.
Minimal avoided patterns are counted by A335465.
Cf. A034691, A056986, A108917, A124767, A124770, A158005, A269134, A333218, A333222, A333224, A334030.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Length[Union[mstype/@Subsets[stc[n]]]],{n,0,30}]

from itertools import combinations
def comp(n):
    # see A357625
    return
def A335465(n):
    A,B,C = set(),set(),comp(n)
    c = range(len(C))
    for j in c:
        for k in combinations(c, j):
            A.add(tuple(C[i] for i in k))
    for i in A:
        D = {v: rank + 1 for rank, v in enumerate(sorted(set(i)))}
        B.add(tuple(D[v] for v in i))
    return len(B)+1 # John Tyler Rascoe, Mar 12 2025

A351014 Number of distinct runs in the n-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 1, 1, 2, 2, 2, 1, 3, 3, 2, 2, 3, 1, 2, 3, 2, 2, 2, 2, 2, 3, 3, 3, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 1, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 2, 2, 3

Offset: 0

Gus Wiseman, Feb 07 2022

nonn

The n-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 n, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The number 3310 has binary expansion 110011101110 and standard composition (1,3,1,1,2,1,1,2), with runs (1), (3), (1,1), (2), (1,1), (2), of which 4 are distinct, so a(3310) = 4.

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

Counting not necessarily distinct runs gives A124767.
Using binary expansions instead of standard compositions gives A297770.
Positions of first appearances are A351015.
A005811 counts runs in binary expansion.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A085207 represents concatenation of standard compositions, reverse A085208.
A333489 ranks anti-runs, complement A348612.
A345167 ranks alternating compositions, counted by A025047.
A351204 counts partitions where every permutation has all distinct runs.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Selected statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
Cf. A098859, A106356, A116608, A238279, A242882, A318928, A325545, A328592, A329745, A350952, A351201.

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

A344619 The a(n)-th composition in standard order (A066099) has alternating sum 0.

Original entry on oeis.org

0, 3, 10, 13, 15, 36, 41, 43, 46, 50, 53, 55, 58, 61, 63, 136, 145, 147, 150, 156, 162, 165, 167, 170, 173, 175, 180, 185, 187, 190, 196, 201, 203, 206, 210, 213, 215, 218, 221, 223, 228, 233, 235, 238, 242, 245, 247, 250, 253, 255, 528, 545, 547, 550, 556, 568

Offset: 1

Gus Wiseman, Jun 03 2021

nonn

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

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 of terms together with the corresponding compositions begins:
    0: ()
    3: (1,1)
   10: (2,2)
   13: (1,2,1)
   15: (1,1,1,1)
   36: (3,3)
   41: (2,3,1)
   43: (2,2,1,1)
   46: (2,1,1,2)
   50: (1,3,2)
   53: (1,2,2,1)
   55: (1,2,1,1,1)
   58: (1,1,2,2)
   61: (1,1,1,2,1)
   63: (1,1,1,1,1,1)
  136: (4,4)
  145: (3,4,1)
  147: (3,3,1,1)
  150: (3,2,1,2)
  156: (3,1,1,3)

The version for Heinz numbers of partitions is A000290, counted by A000041.
These are the positions of zeros in A344618.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A116406 counts compositions with alternating sum >= 0.
A124754 gives the alternating sum of standard compositions.
A316524 is the alternating sum of the prime indices of n.
A344604 counts wiggly compositions with twins.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
A344616 gives the alternating sum of reversed prime indices.
All of the following pertain to compositions in standard order:
- The length is A000120.
- Converting to reversed ranking gives A059893.
- The rows are A066099.
- The sum is A070939.
- The runs are counted by A124767.
- The reversed version is A228351.
- Strict compositions are ranked by A233564.
- Constant compositions are ranked by A272919.
- The Heinz number is A333219.
- Anti-run compositions are ranked by A333489.
Cf. A000070, A000097, A003242, A006330, A028260, A119899, A239830, A344605, A344607, A344650, A344739.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]]
Select[Range[0,100],ats[stc[#]]==0&]

A053754 If k is in the sequence then 2k and 2k+1 are not (and 0 is in the sequence); when written in binary k has an even number of bits (0 has 0 digits).

Original entry on oeis.org

0, 2, 3, 8, 9, 10, 11, 12, 13, 14, 15, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148

Offset: 1

Henry Bottomley, Apr 06 2000

Runs of successive terms with same number of bits have length twice powers of 4 (A081294). [Clarified by Michel Marcus, Oct 21 2020]
The sequence A081294 counts compositions of even numbers - Gus Wiseman, Aug 12 2021
A031443 is a subsequence; A179888 is the intersection of this sequence and A032925. - Reinhard Zumkeller, Jul 31 2010
The lower and upper asymptotic densities of this sequence are 1/3 and 2/3, respectively. - Amiram Eldar, Feb 01 2021
From Gus Wiseman, Aug 10 2021: (Start)
Also numbers k such that the k-th composition in standard order (row k of A066099) has even sum. The terms and corresponding compositions begin:
  0: ()   2: (2)      8: (4)
          3: (1,1)    9: (3,1)
                     10: (2,2)
                     11: (2,1,1)
                     12: (1,3)
                     13: (1,2,1)
                     14: (1,1,2)
                     15: (1,1,1,1)
The following pertain to compositions in standard order: A000120, A029837, A070939, A066099, A124767.
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10001

Cf. A031443, A032925, A179888.
Positions of even terms in A029837 with offset 0.
The complement (the odd version) is A053738, counted by A000302.
The version for Heinz numbers of partitions is A300061, counted by A058696.
Cf. A000120, A001969, A008549, A070939, A080791, A114121.

a053754 n = a053754_list !! (n-1)
a053754_list = 0 : filter (even . a070939) [1..]
-- Reinhard Zumkeller, Apr 18 2015

Select[Range[0, 150], EvenQ @ IntegerLength[#, 2] &] (* Amiram Eldar, Feb 01 2021 *)

lista(nn) = {my(va = vector(nn)); for (n=2, nn, my(k=va[n-1]+1); while (#select(x->(x==k\2), va), k++); va[n] = k;); va;} \\ Michel Marcus, Oct 20 2020

PARI
```
a(n) = n-1 + (1<Kevin Ryde, Apr 30 2021
```

Offset corrected by Reinhard Zumkeller, Jul 30 2010

cp's OEIS Frontend

A333627 The a(n)-th composition in standard order is the sequence of run-lengths of the n-th composition in standard order.

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A124765 Number of monotonically decreasing runs for compositions in standard order.

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A335465 Number of minimal normal patterns avoided by the n-th composition in standard order (A066099).

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A373948 Run-compression encoded as a transformation of compositions in standard order.

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A124769 Number of strictly decreasing runs for compositions in standard order.

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A344618 Reverse-alternating sums of standard compositions (A066099). Alternating sums of the compositions ranked by A228351.

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A335454 Number of normal patterns matched by the n-th composition in standard order (A066099).

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A351014 Number of distinct runs in the n-th composition in standard order.

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A053754 If k is in the sequence then 2k and 2k+1 are not (and 0 is in the sequence); when written in binary k has an even number of bits (0 has 0 digits).