A335456 -id:A335456 - cp's OEIS frontend

A335488 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,1).

3, 7, 10, 11, 13, 14, 15, 19, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 35, 36, 39, 42, 43, 45, 46, 47, 49, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 97, 99, 100, 101

Offset: 1

Gus Wiseman, Jun 18 2020

nonn

These are compositions with some part appearing more than once, or non-strict compositions.
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:
   3: (1,1)
   7: (1,1,1)
  10: (2,2)
  11: (2,1,1)
  13: (1,2,1)
  14: (1,1,2)
  15: (1,1,1,1)
  19: (3,1,1)
  21: (2,2,1)
  22: (2,1,2)
  23: (2,1,1,1)
  25: (1,3,1)
  26: (1,2,2)
  27: (1,2,1,1)
  28: (1,1,3)

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 A233564 is the avoiding version.
Patterns matching this pattern are counted by A019472 (by length).
Permutations of prime indices matching this pattern are counted by A335487.
These compositions are counted by A261982 (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.
The (1,1,1)-matching case is A335512.
Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A334968, A335456, A335458.

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

A335508 Number of patterns of length n matching the pattern (1,1,1).

Original entry on oeis.org

0, 0, 0, 1, 9, 91, 993, 12013, 160275, 2347141, 37496163, 649660573, 12142311195, 243626199181, 5224710549243, 119294328993853, 2889836999693355, 74037381200415901, 2000383612949821323, 56850708386783835133, 1695491518035158123115, 52949018580275965241821

Offset: 0

Gus Wiseman, Jun 18 2020

nonn

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 a(3) = 1 through a(4) = 9 patterns:
  (1,1,1)  (1,1,1,1)
           (1,1,1,2)
           (1,1,2,1)
           (1,2,1,1)
           (1,2,2,2)
           (2,1,1,1)
           (2,1,2,2)
           (2,2,1,2)
           (2,2,2,1)

Alois P. Heinz, Table of n, a(n) for n = 0..424
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The complement A080599 is the avoiding version.
Permutations of prime indices matching this pattern are counted by A335510.
Compositions matching this pattern are counted by A335455 and ranked by A335512.
Patterns are counted by A000670 and ranked by A333217.
Patterns matching the pattern (1,1) are counted by A019472.
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.
Patterns matching (1,2,3) are counted by A335515.
Cf. A034691, A056986, A232464, A238279, A292884, A333175, A333755, A335451, A335456, A335457, A335458.
Cf. A276922.

b:= proc(n, k) option remember; `if`(n=0, 1, add(
      b(n-i, k)*binomial(n, i), i=1..min(n, k)))
    end:
a:= n-> b(n$2)-b(n, 2):
seq(a(n), n=0..21);  # Alois P. Heinz, Jan 28 2024

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],MatchQ[#,{_,x_,_,x_,_,x_,_}]&]],{n,0,6}]

a(n) = Sum_{k=3..n} A276922(n,k). - Alois P. Heinz, Jan 28 2024

a(n) = A000670(n) - A080599(n). - Andrew Howroyd, Jan 28 2024

a(9)-a(21) from Alois P. Heinz, Jan 28 2024

A335510 Number of (1,1,1)-matching permutations of the prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 5, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jun 19 2020

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

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.

Patterns matching this pattern are counted by A335508.
These compositions are counted by A335455.
The (1,1)-matching version is A335487.
The complement A335511 is the avoiding version.
These permutations are ranked by A335512.
Permutations of prime indices are counted by A008480.
Patterns are counted by A000670 and ranked by A333217.
Anti-run permutations of prime indices are counted by A335452.
Cf. A056239, A056986, A112798, A238279, A281188, A333221, A333755, A335456, A335460, A335462, A335463.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],MatchQ[#,{_,x_,_,x_,_,x_,_}]&]],{n,0,100}]

If n is cubefree, a(n) = 0; otherwise a(n) = A008480(n).

A335512 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,1,1).

Original entry on oeis.org

7, 15, 23, 27, 29, 30, 31, 39, 42, 47, 51, 55, 57, 59, 60, 61, 62, 63, 71, 79, 85, 86, 87, 90, 91, 93, 94, 95, 99, 103, 106, 107, 109, 110, 111, 113, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 135, 143, 151, 155, 157, 158, 159, 167, 170, 171

Offset: 1

Gus Wiseman, Jun 18 2020

nonn

These are compositions with some part appearing more than twice.
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:
   7: (1,1,1)
  15: (1,1,1,1)
  23: (2,1,1,1)
  27: (1,2,1,1)
  29: (1,1,2,1)
  30: (1,1,1,2)
  31: (1,1,1,1,1)
  39: (3,1,1,1)
  42: (2,2,2)
  47: (2,1,1,1,1)
  51: (1,3,1,1)
  55: (1,2,1,1,1)
  57: (1,1,3,1)
  59: (1,1,2,1,1)
  60: (1,1,1,3)

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 A335513 is the avoiding version.
Patterns matching this pattern are counted by A335508 (by length).
Permutations of prime indices matching this pattern are counted by A335510.
These compositions are counted by A335455 (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.
The (1,1)-matching version is A335488.
Cf. A034691, A056986, A114994, A238279, A334968, A335456, A335458.

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

A335521 Number of (1,2,3)-avoiding permutations of the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 5, 1, 1, 2, 2, 2, 6, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 1, 2, 5, 1, 3, 2, 5, 1, 10, 1, 2, 3, 3, 2, 5, 1, 5, 1, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Jun 19 2020

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

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 a(n) permutations for n = 1, 6, 12, 24, 30, 36, 60, 72, 120:
  ()  (12)  (112)  (1112)  (132)  (1122)  (1132)  (11122)  (11132)
      (21)  (121)  (1121)  (213)  (1212)  (1312)  (11212)  (11312)
            (211)  (1211)  (231)  (1221)  (1321)  (11221)  (11321)
                   (2111)  (312)  (2112)  (2113)  (12112)  (13112)
                           (321)  (2121)  (2131)  (12121)  (13121)
                                  (2211)  (2311)  (12211)  (13211)
                                          (3112)  (21112)  (21113)
                                          (3121)  (21121)  (21131)
                                          (3211)  (21211)  (21311)
                                                  (22111)  (23111)
                                                           (31112)
                                                           (31121)
                                                           (31211)
                                                           (32111)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

These compositions are counted by A102726.
Patterns avoiding this pattern are counted by A226316.
The complement A335520 is the matching version.
Permutations of prime indices are counted by A008480.
Patterns are counted by A000670 and ranked by A333217.
Anti-run permutations of prime indices are counted by A335452.
Cf. A056239, A056986, A112798, A238279, A281188, A333221, A333755, A335456, A335460, A335462, A335463.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],!MatchQ[#,{_,x_,_,y_,_,z_,_}/;x

For n > 0, a(n) + A335520(n) = A008480(n).

A374766 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of maximal strictly decreasing runs sum to k.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 0, 3, 5, 0, 0, 0, 1, 8, 7, 0, 0, 0, 1, 3, 17, 11, 0, 0, 0, 0, 4, 10, 35, 15, 0, 0, 0, 0, 1, 12, 28, 65, 22, 0, 0, 0, 0, 1, 6, 31, 70, 118, 30, 0, 0, 0, 0, 1, 3, 22, 78, 163, 203, 42, 0, 0, 0, 0, 0, 4, 13, 69, 186, 354, 342, 56

Offset: 0

Gus Wiseman, Aug 02 2024

The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.

Are the column-sums finite?

			Triangle begins:
   1
   0   1
   0   0   2
   0   0   1   3
   0   0   0   3   5
   0   0   0   1   8   7
   0   0   0   1   3  17  11
   0   0   0   0   4  10  35  15
   0   0   0   0   1  12  28  65  22
   0   0   0   0   1   6  31  70 118  30
   0   0   0   0   1   3  22  78 163 203  42
   0   0   0   0   0   4  13  69 186 354 342  56
Row n = 6 counts the following compositions:
  .  .  .  (321)  (42)    (51)     (6)
                  (132)   (411)    (15)
                  (2121)  (141)    (24)
                          (312)    (114)
                          (231)    (33)
                          (213)    (123)
                          (3111)   (1113)
                          (1311)   (222)
                          (1131)   (1122)
                          (2211)   (11112)
                          (2112)   (111111)
                          (1221)
                          (1212)
                          (21111)
                          (12111)
                          (11211)
                          (11121)

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

Column n = k is A000041.
Row-sums are A011782.
For length instead of sum we have A333213.
The corresponding rank statistic is A374758, row-sums of A374757.
For identical leaders we have A374760, ranks A374759.
For distinct leaders we have A374761, ranks A374767.
Other types of runs (instead of strictly decreasing):
- For leaders of identical runs we have A373949.
- For leaders of anti-runs we have A374521.
- For leaders of weakly increasing runs we have A374637.
- For leaders of strictly increasing runs we have A374700.
- For leaders of weakly decreasing runs we have A374748.
A003242 counts anti-run compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
Cf. A106356, A238343, A261982, A274174, A374517, A374518, A374687.

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

A375135 Number of integer compositions of n whose leaders of maximal strictly increasing runs are not weakly decreasing.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 3, 9, 25, 63, 152, 355, 809, 1804, 3963, 8590, 18423, 39161, 82620, 173198, 361101, 749326, 1548609, 3189132, 6547190, 13404613, 27378579, 55801506, 113517749, 230544752, 467519136, 946815630, 1915199736, 3869892105, 7812086380, 15756526347

Offset: 0

Gus Wiseman, Aug 06 2024

nonn

The leaders of maximal strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.

			The composition y = (1,2,1,3,2,3) has strictly increasing runs ((1,2),(1,3),(2,3)), with leaders (1,1,2), which are not weakly decreasing, so y is counted under a(12).
The a(0) = 0 through a(8) = 25 compositions:
  .  .  .  .  .  (122)  (132)   (133)    (143)
                        (1122)  (142)    (152)
                        (1221)  (1132)   (233)
                                (1222)   (1133)
                                (1321)   (1142)
                                (2122)   (1223)
                                (11122)  (1232)
                                (11221)  (1322)
                                (12211)  (1331)
                                         (1421)
                                         (2132)
                                         (3122)
                                         (11132)
                                         (11222)
                                         (11321)
                                         (12122)
                                         (12212)
                                         (12221)
                                         (13211)
                                         (21122)
                                         (21221)
                                         (111122)
                                         (111221)
                                         (112211)
                                         (122111)

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

For leaders of constant runs we have A056823.
For leaders of weakly increasing runs we have A374636, complement A189076?
The complement is counted by A374697.
For leaders of anti-runs we have A374699, complement A374682.
Other functional neighbors: A188920, A374764, A374765.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
A373949 counts compositions by run-compressed sum, opposite A373951.
A374700 counts compositions by sum of leaders of strictly increasing runs.
Cf. A106356, A238343, A261982, A333213, A374632, A374679, A374683, A374689.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], !GreaterEqual@@First/@Split[#,Less]&]],{n,0,15}]

a(n) = A011782(n) - A374697(n). - Jinyuan Wang, Feb 13 2025

More terms from Jinyuan Wang, Feb 13 2025

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

A335513 Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (1,1,1).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 32, 33, 34, 35, 36, 37, 38, 40, 41, 43, 44, 45, 46, 48, 49, 50, 52, 53, 54, 56, 58, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 88, 89

Offset: 1

Gus Wiseman, Jun 18 2020

nonn

These are compositions with no part appearing more than twice.
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:
   0: ()         17: (4,1)      37: (3,2,1)
   1: (1)        18: (3,2)      38: (3,1,2)
   2: (2)        19: (3,1,1)    40: (2,4)
   3: (1,1)      20: (2,3)      41: (2,3,1)
   4: (3)        21: (2,2,1)    43: (2,2,1,1)
   5: (2,1)      22: (2,1,2)    44: (2,1,3)
   6: (1,2)      24: (1,4)      45: (2,1,2,1)
   8: (4)        25: (1,3,1)    46: (2,1,1,2)
   9: (3,1)      26: (1,2,2)    48: (1,5)
  10: (2,2)      28: (1,1,3)    49: (1,4,1)
  11: (2,1,1)    32: (6)        50: (1,3,2)
  12: (1,3)      33: (5,1)      52: (1,2,3)
  13: (1,2,1)    34: (4,2)      53: (1,2,2,1)
  14: (1,1,2)    35: (4,1,1)    54: (1,2,1,2)
  16: (5)        36: (3,3)      56: (1,1,4)

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

These compositions are counted by A232432 (by sum).
The (1,1)-avoiding version is A233564.
The complement A335512 is the matching version.
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.
Patterns avoiding (1,1,1) are counted by A080599 (by length).
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.
Permutations of prime indices avoiding (1,1,1) are counted by A335511.
Cf. A034691, A056986, A108917, A114994, A238279, A334968, A335456, A335458.

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

cp's OEIS Frontend

A335488 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,1).

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A335508 Number of patterns of length n matching the pattern (1,1,1).

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A335510 Number of (1,1,1)-matching permutations of the prime indices of n.

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A335512 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,1,1).

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A335521 Number of (1,2,3)-avoiding permutations of the prime indices of n.

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A374766 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of maximal strictly decreasing runs sum to k.

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A375135 Number of integer compositions of n whose leaders of maximal strictly increasing runs are not weakly decreasing.

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

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