A333224 -id:A333224 - cp's OEIS frontend

A334300 Number of distinct nonempty subsequences (not necessarily contiguous) in the n-th composition in standard order (A066099).

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

Offset: 0

Gus Wiseman, Jun 01 2020

Looking only at contiguous subsequences, or restrictions to a subinterval, gives A124770.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			Triangle begins:
  1
  1 2
  1 3 3 3
  1 3 2 5 3 6 5 4
  1 3 3 5 3 5 6 7 3 6 5 9 5 9 7 5
If the k-th composition in standard order is c, then we say that the STC-number of c is k. The n-th column below lists the STC-numbers of the nonempty subsequences of the composition with STC-number n:
  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15
        1     2  2  3     4   2   5   4   6   6   7
              1  1  1     1       3   1   5   3   3
                                  2       3   2   1
                                  1       2   1
                                          1

John Tyler Rascoe, Table of n, a(n) for n = 0..8192

Row lengths are A011782.
Looking only at contiguous subsequences gives A124770.
The contiguous case with empty subsequences allowed is A124771.
Allowing empty subsequences gives A334299.
Compositions where every subinterval has a different sum are A333222.
Knapsack compositions are A333223.
Contiguous positive subsequence-sums are counted by A333224.
Contiguous subsequence-sums are counted by A333257.
Subsequence-sums are counted by A334968.
Cf. A000120, A029931, A048793, A066099, A070939, A108917, A325676, A334967.

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

from itertools import combinations
def comp(n):
    # see A357625
    return
def A334300(n):
    A,C = 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))
    return len(A) # John Tyler Rascoe, Mar 12 2025

a(n) = A334299(n) - 1.

A335474 Number of nonempty normal patterns contiguously matched by the n-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 4, 2, 4, 4, 4, 1, 2, 2, 4, 2, 4, 4, 6, 2, 4, 4, 7, 4, 7, 6, 5, 1, 2, 2, 4, 2, 3, 4, 6, 2, 4, 3, 6, 4, 6, 7, 8, 2, 4, 4, 7, 3, 7, 6, 10, 4, 7, 6, 10, 6, 10, 8, 6, 1, 2, 2, 4, 2, 3, 4, 6, 2, 4, 4, 6, 4, 6, 7, 8, 2, 4, 4, 7, 4, 6

Offset: 0

Gus Wiseman, Jun 21 2020

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 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 a(n) patterns for n = 32, 80, 133, 290, 305, 329, 436 are:
      (1)  (1)   (1)    (1)    (1)     (1)     (1)
           (12)  (21)   (12)   (12)    (11)    (12)
                 (321)  (21)   (21)    (12)    (21)
                        (231)  (121)   (21)    (121)
                               (213)   (122)   (123)
                               (2131)  (221)   (212)
                                       (2331)  (1212)
                                               (2123)
                                               (12123)

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

The version for Heinz numbers of partitions is A335516(n) - 1.
The non-contiguous version is A335454(n) - 1.
The version allowing empty patterns is A335458.
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 A334299(n) distinct subsequences.
Minimal avoided patterns are counted by A335465.
Patterns matched by prime indices are counted by A335549.
Cf. A034691, A056986, A108917, A124767, A124770, A181796, A269134, A333222, A333224, A335456, A335457.

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/@ReplaceList[stc[n],{_,s__,_}:>{s}]]],{n,0,100}]

a(n) = A335458(n) - 1.

A335475 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,2,2).

Original entry on oeis.org

26, 53, 54, 58, 90, 100, 106, 107, 109, 110, 117, 118, 122, 154, 164, 181, 182, 186, 201, 202, 204, 210, 212, 213, 214, 215, 218, 219, 221, 222, 228, 234, 235, 237, 238, 245, 246, 250, 282, 309, 310, 314, 329, 332, 346, 356, 362, 363, 365, 366, 373, 374, 378

Offset: 1

Gus Wiseman, Jun 18 2020

nonn

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.

We define a 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 sequence of terms together with the corresponding compositions begins:
   26: (1,2,2)
   53: (1,2,2,1)
   54: (1,2,1,2)
   58: (1,1,2,2)
   90: (2,1,2,2)
  100: (1,3,3)
  106: (1,2,2,2)
  107: (1,2,2,1,1)
  109: (1,2,1,2,1)
  110: (1,2,1,1,2)
  117: (1,1,2,2,1)
  118: (1,1,2,1,2)
  122: (1,1,1,2,2)
  154: (3,1,2,2)
  164: (2,3,3)

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.

The complement A335525 is the avoiding version.
The (2,2,1)-matching version is A335477.
Patterns matching this pattern are counted by A335509 (by length).
Permutations of prime indices matching this pattern are counted by A335453.
These compositions are counted by A335472 (by sum).
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335446, A335456, A335458.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,y_,_,y_,_}/;x

A335476 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,1,2).

Original entry on oeis.org

14, 28, 29, 30, 46, 54, 56, 57, 58, 59, 60, 61, 62, 78, 84, 92, 93, 94, 102, 108, 109, 110, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 142, 156, 157, 158, 168, 169, 172, 174, 180, 182, 184, 185, 186, 187, 188, 189, 190, 198, 204

Offset: 1

Gus Wiseman, Jun 18 2020

nonn

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.

We define a 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 sequence of terms together with the corresponding compositions begins:
  14: (1,1,2)
  28: (1,1,3)
  29: (1,1,2,1)
  30: (1,1,1,2)
  46: (2,1,1,2)
  54: (1,2,1,2)
  56: (1,1,4)
  57: (1,1,3,1)
  58: (1,1,2,2)
  59: (1,1,2,1,1)
  60: (1,1,1,3)
  61: (1,1,1,2,1)
  62: (1,1,1,1,2)
  78: (3,1,1,2)
  84: (2,2,3)

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.

The complement A335522 is the avoiding version.
The (2,1,1)-matching version is A335478.
Patterns matching this pattern are counted by A335509 (by length).
Permutations of prime indices matching this pattern are counted by A335446.
These compositions are counted by A335470 (by sum).
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335446, A335456, A335458.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,x_,_,y_,_}/;x

A335477 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (2,2,1).

Original entry on oeis.org

21, 43, 45, 53, 73, 85, 86, 87, 91, 93, 107, 109, 117, 146, 147, 149, 153, 165, 169, 171, 172, 173, 174, 175, 181, 182, 183, 187, 189, 201, 213, 214, 215, 219, 221, 235, 237, 245, 273, 277, 293, 294, 295, 297, 299, 301, 306, 307, 309, 313, 325, 329, 331, 333

Offset: 1

Gus Wiseman, Jun 18 2020

nonn

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.

We define a 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 sequence of terms together with the corresponding compositions begins:
   21: (2,2,1)
   43: (2,2,1,1)
   45: (2,1,2,1)
   53: (1,2,2,1)
   73: (3,3,1)
   85: (2,2,2,1)
   86: (2,2,1,2)
   87: (2,2,1,1,1)
   91: (2,1,2,1,1)
   93: (2,1,1,2,1)
  107: (1,2,2,1,1)
  109: (1,2,1,2,1)
  117: (1,1,2,2,1)
  146: (3,3,2)
  147: (3,3,1,1)

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

The complement A335524 is the avoiding version.
The (1,2,2)-matching version is A335475.
Patterns matching this pattern are counted by A335509 (by length).
Permutations of prime indices matching this pattern are counted by A335453.
These compositions are counted by A335472 (by sum).
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335446, A335456, A335458.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,x_,_,y_,_}/;x>y]&]

A334268 Number of compositions of n where every distinct subsequence (not necessarily contiguous) has a different sum.

Original entry on oeis.org

1, 1, 2, 4, 5, 10, 10, 24, 24, 43, 42, 88, 72, 136, 122, 242, 213, 392, 320, 630, 490, 916, 742, 1432, 1160, 1955, 1604, 2826, 2310, 3850, 2888, 5416, 4426, 7332, 5814, 10046, 7983, 12946, 10236, 17780, 14100, 22674, 17582, 30232, 23674, 37522, 29426, 49832

Offset: 0

Gus Wiseman, Jun 02 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

The contiguous case is A325676.

			The a(1) = 1 through a(6) = 19 compositions:
  (1)  (2)    (3)      (4)        (5)          (6)
       (1,1)  (1,2)    (1,3)      (1,4)        (1,5)
              (2,1)    (2,2)      (2,3)        (2,4)
              (1,1,1)  (3,1)      (3,2)        (3,3)
                       (1,1,1,1)  (4,1)        (4,2)
                                  (1,1,3)      (5,1)
                                  (1,2,2)      (1,1,4)
                                  (2,2,1)      (2,2,2)
                                  (3,1,1)      (4,1,1)
                                  (1,1,1,1,1)  (1,1,1,1,1,1)

These compositions are ranked by A334967.
Compositions where every restriction to a subinterval has a different sum are counted by A169942 and A325677 and ranked by A333222. The case of partitions is counted by A325768 and ranked by A325779.
Positive subset-sums of partitions are counted by A276024 and A299701.
Knapsack partitions are counted by A108917 and A325592 and ranked by A299702, while the strict case is counted by A275972 and ranked by A059519 and A301899.
Knapsack compositions are counted by A325676 and A325687 and ranked by A333223. The case of partitions is counted by A325769 and ranked by A325778, for which the number of distinct consecutive subsequences is given by A325770.
Cf. A003022, A029931, A103295, A143823, A325680, A333224.

b:= proc(n, s) option remember; `if`(n=0, 1, add((h->
      `if`(nops(h)=nops(map(l-> add(i, i=l), h)),
       b(n-j, h), 0))({s[], map(l-> [l[], j], s)[]}), j=1..n))
    end:
a:= n-> b(n, {[]}):
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 03 2020

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Total/@Union[Subsets[#]]&]],{n,0,15}]

a(18)-a(47) from Alois P. Heinz, Jun 03 2020

A335478 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (2,1,1).

Original entry on oeis.org

11, 19, 23, 27, 35, 39, 43, 45, 46, 47, 51, 55, 59, 67, 71, 74, 75, 77, 78, 79, 83, 87, 89, 91, 92, 93, 94, 95, 99, 103, 107, 109, 110, 111, 115, 119, 123, 131, 135, 138, 139, 141, 142, 143, 147, 149, 150, 151, 153, 154, 155, 156, 157, 158, 159, 163, 167, 171

Offset: 1

Gus Wiseman, Jun 18 2020

nonn

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.

We define a 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 sequence of terms together with the corresponding compositions begins:
  11: (2,1,1)
  19: (3,1,1)
  23: (2,1,1,1)
  27: (1,2,1,1)
  35: (4,1,1)
  39: (3,1,1,1)
  43: (2,2,1,1)
  45: (2,1,2,1)
  46: (2,1,1,2)
  47: (2,1,1,1,1)
  51: (1,3,1,1)
  55: (1,2,1,1,1)
  59: (1,1,2,1,1)
  67: (5,1,1)
  71: (4,1,1,1)

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

The complement A335523 is the avoiding version.
The (1,1,2)-matching version is A335476.
Patterns matching this pattern are counted by A335509 (by length).
Permutations of prime indices matching this pattern are counted by A335516.
These compositions are counted by A335470 (by sum).
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335446, A335456, A335458, A335475.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,y_,_,y_,_}/;x>y]&]

A335522 Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (1,1,2).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 55, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81

Offset: 1

Gus Wiseman, Jun 18 2020

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 a 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).

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 avoiding this pattern are counted by A001710 (by length).
Permutations of prime indices avoiding this pattern are counted by A335449.
These compositions are counted by A335471 (by sum).
The complement A335476 is the matching version.
The (2,1,1)-avoiding version is A335523.
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335446, A335456, A335458.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],!MatchQ[stc[#],{_,x_,_,x_,_,y_,_}/;x

A335523 Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (2,1,1).

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, 26, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 40, 41, 42, 44, 48, 49, 50, 52, 53, 54, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 72, 73, 76, 80, 81, 82, 84, 85, 86, 88, 90

Offset: 1

Gus Wiseman, Jun 18 2020

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 a 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).

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 avoiding this pattern are counted by A001710 (by length).
Permutations of prime indices avoiding this pattern are counted by A335449.
These compositions are counted by A335471 (by sum).
The complement A335478 is the matching version.
The (1,1,2)-avoiding version is A335522.
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335446, A335456, A335458.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],!MatchQ[stc[#],{_,x_,_,y_,_,y_,_}/;x>y]&]

A353696 Numbers k such that the k-th composition in standard order (A066099) is empty, a singleton, or has run-lengths that are a consecutive subsequence 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 22 2022

nonn

First differs from the non-consecutive version A353431 in lacking 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 a 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 terms together with their corresponding compositions begin:
    0: ()
    1: (1)
    2: (2)
    4: (3)
    8: (4)
   10: (2,2)
   16: (5)
   32: (6)
   43: (2,2,1,1)
   58: (1,1,2,2)
   64: (7)
  128: (8)
  256: (9)
  292: (3,3,3)
  349: (2,2,1,1,2,1)
  442: (1,2,1,1,2,2)
  512: (10)
  586: (3,3,2,2)
  676: (2,2,3,3)
  697: (2,2,1,1,3,1)
  826: (1,3,1,1,2,2)

Non-recursive non-consecutive for partitions: A325755, counted by A325702.
Non-consecutive: A353431, counted by A353391.
Non-consecutive for partitions: A353393, counted by A353426.
Non-recursive non-consecutive: A353402, counted by A353390.
Counted by: A353430.
Non-recursive: A353432, counted by A353392.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order, run-lengths A333769.
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, multisets A225620, sets A333256.
- Runs are A272919, counted by A000005.
- Golomb rulers are A333222, counted by A169942.
- Anti-runs are A333489, counted by A003242.
Cf. A032020, A114640, A181819, A228351, A329739, A318928, A325705, A329738, A333224, A353427, A353403.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
yoyQ[y_]:=Length[y]<=1||MemberQ[Join@@Table[Take[y,{i,j}],{i,Length[y]},{j,i,Length[y]}],Length/@Split[y]]&&yoyQ[Length/@Split[y]];
Select[Range[0,1000],yoyQ[stc[#]]&]

cp's OEIS Frontend

A334300 Number of distinct nonempty subsequences (not necessarily contiguous) in the n-th composition in standard order (A066099).

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A335474 Number of nonempty normal patterns contiguously matched by the n-th composition in standard order.

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A335475 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,2,2).

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A335476 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,1,2).

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A335477 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (2,2,1).

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A334268 Number of compositions of n where every distinct subsequence (not necessarily contiguous) has a different sum.

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A335478 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (2,1,1).

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A335522 Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (1,1,2).

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