A066099 -id:A066099 - cp's OEIS frontend

A375295 Numbers k such that the leaders of maximal weakly increasing runs in the k-th composition in standard order (row k of A066099) are not strictly decreasing.

13, 25, 27, 29, 45, 49, 50, 51, 53, 54, 55, 57, 59, 61, 77, 82, 89, 91, 93, 97, 98, 99, 101, 102, 103, 105, 107, 108, 109, 110, 111, 113, 114, 115, 117, 118, 119, 121, 123, 125, 141, 153, 155, 157, 162, 165, 173, 177, 178, 179, 181, 182, 183, 185, 187, 189

Offset: 1

Gus Wiseman, Aug 12 2024

nonn

First differs from the non-dashed version in lacking 166, corresponding to the composition (2,3,1,2).
The leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.
Also numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed patterns 1-32 or 1-21.

			The sequence together with corresponding compositions begins:
  13: (1,2,1)
  25: (1,3,1)
  27: (1,2,1,1)
  29: (1,1,2,1)
  45: (2,1,2,1)
  49: (1,4,1)
  50: (1,3,2)
  51: (1,3,1,1)
  53: (1,2,2,1)
  54: (1,2,1,2)
  55: (1,2,1,1,1)
  57: (1,1,3,1)
  59: (1,1,2,1,1)
  61: (1,1,1,2,1)
  77: (3,1,2,1)
  82: (2,3,2)
  89: (2,1,3,1)
  91: (2,1,2,1,1)
  93: (2,1,1,2,1)

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

For leaders of identical runs we have A335485.
Positions of non-strictly decreasing rows in A374629 (sums A374630).
For identical leaders we have A374633, counted by A374631.
Matching 1-32 only gives A375137, reverse A375138, both counted by A374636.
Interchanging weak/strict gives A375139, counted by A375135.
Compositions of this type are counted by A375140, complement A188920.
The reverse version is A375296.
A003242 counts anti-runs, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A374637 counts compositions by sum of leaders of weakly increasing runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Leader is A065120.
- Parts are listed by A066099, reverse A228351.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Run-length transform is A333627, sum A070939.
- Run-counts: A124766, A124765, A124768, A124769, A333381, A124767.
- Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.
Cf. A056823, A106356, A188919, A189076, A238343, A261982, A333213, A335480, A335482, A373948, A374746, A374768, A375123.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!Greater@@First/@Split[stc[#],LessEqual]&]
- or -
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,z_,y_,_}/;x<=y

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

Original entry on oeis.org

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.

A335374 Numbers k such that the k-th composition in standard order (A066099) is not co-unimodal.

Original entry on oeis.org

13, 25, 27, 29, 41, 45, 49, 50, 51, 53, 54, 55, 57, 59, 61, 77, 81, 82, 83, 89, 91, 93, 97, 98, 99, 101, 102, 103, 105, 107, 108, 109, 110, 111, 113, 114, 115, 117, 118, 119, 121, 123, 125, 141, 145, 153, 155, 157, 161, 162, 163, 165, 166, 167, 169, 173, 177

Offset: 1

Gus Wiseman, Jun 03 2020

nonn

A sequence of integers is co-unimodal if it is the concatenation of a weakly decreasing and a weakly increasing sequence, implying that its negation is unimodal.

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 together with the corresponding compositions begins:
  13: (1,2,1)
  25: (1,3,1)
  27: (1,2,1,1)
  29: (1,1,2,1)
  41: (2,3,1)
  45: (2,1,2,1)
  49: (1,4,1)
  50: (1,3,2)
  51: (1,3,1,1)
  53: (1,2,2,1)
  54: (1,2,1,2)
  55: (1,2,1,1,1)
  57: (1,1,3,1)
  59: (1,1,2,1,1)
  61: (1,1,1,2,1)
  77: (3,1,2,1)
  81: (2,4,1)
  82: (2,3,2)
  83: (2,3,1,1)
  89: (2,1,3,1)

This is the dual version of A335373.
The case that is not unimodal either is A335375.
Unimodal compositions are A001523.
Unimodal normal sequences are A007052.
Unimodal permutations are A011782.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Numbers with non-unimodal unsorted prime signature are A332282.
Co-unimodal compositions are A332578.
Numbers with non-co-unimodal unsorted prime signature are A332642.
Non-co-unimodal compositions are A332669.
Cf. A000120, A029931, A048793, A066099, A070939, A334299.
Cf. A112798, A227038, A329398, A332281, A332286, A332287, A332638, A332639, A332643, A332670, A332873, A333146.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!unimodQ[-stc[#]]&]

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_,_}]&]

A276165 a(n) is the first-player score difference of a "Coins in a Row" game over the n-th row of A066099 using a minimax strategy.

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

Offset: 0

Peter Kagey, Aug 25 2016

sign

"Coins in a Row" is a game in which players alternate picking up coins of varying denominations from the end of the row in an attempt to collect as many points as possible.

When a(n) is negative, the second player has a strategy that is guaranteed to collect more points.

			Let [R,L,L,L] represent a game in which the first player takes the right coin, the second player takes the left coin, the first player takes the left coin, and the second player takes the left (only remaining) coin.
A066099_Row(0)    = [0];         a(0)    = 0  via [L]
A066099_Row(1)    = [1];         a(1)    = 1  via [L]
A066099_Row(3)    = [1,1];       a(3)    = 0  via [R,L]
A066099_Row(22)   = [2,1,2];     a(22)   = 1  via [L,R,L]
A066099_Row(88)   = [2,1,4];     a(88)   = 3  via [R,L,L]
A066099_Row(1418) = [2,1,4,2,2]; a(1418) = -1 via [L,R,R,R,L]

Peter Winkler, Mathematical Puzzles: A Connoisseur's Collection, A K Peters/CRC Press, 2003, pages 1-2.

Peter Kagey, Table of n, a(n) for n = 0..10000

Cf. A276163, A276164, A276166, A276167.

minimax [] = 0
minimax as = max (head as - minimax (tail as)) (last as - minimax (init as))
a276165 = minimax . a066099_row

a(n) = A276166(n) - A276167(n).

A276166 a(n) is the first player's score in a "Coins in a Row" game over the n-th row of A066099 using a minimax strategy.

Original entry on oeis.org

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

Offset: 0

Peter Kagey, Aug 25 2016

nonn

"Coins in a Row" is a game in which players alternate picking up coins of varying denominations from the end of the row in an attempt to collect as many points as possible.

			Let [R,L,L,L] represent a game in which the first player takes the right coin, the second player takes the left coin, the first player takes the left coin, and the second player takes the left (only remaining) coin.
A066099_Row(0)    = [0];         a(0)    = 0 via [L]
A066099_Row(1)    = [1];         a(1)    = 1 via [L]
A066099_Row(3)    = [1,1];       a(3)    = 1 via [R,L]
A066099_Row(22)   = [2,1,2];     a(22)   = 3 via [L,R,L]
A066099_Row(88)   = [2,1,4];     a(88)   = 5 via [R,L,L]
A066099_Row(1418) = [2,1,4,2,2]; a(1418) = 5 via [L,R,R,R,L]

Peter Winkler, Mathematical Puzzles: A Connoisseur's Collection, A K Peters/CRC Press, 2003, pages 1-2.

Peter Kagey, Table of n, a(n) for n = 0..10000

Cf. A276163, A276164, A276165, A276167.

minimaxDifference [] = 0
minimaxDifference as = max (head as - minimaxDifference (tail as)) (last as - minimaxDifference (init as))
minimaxScore as = (sum as + minimaxDifference as) `div` 2
a276166 = minimaxScore . a066099_row

a(n) = A029837(n + 1) - A276167(n).

a(n) = A276165(n) + A276167(n).

A276167 a(n) is the second player's score in a "Coins in a Row" game over the n-th row of A066099 using a minimax strategy.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 2, 1, 2, 0, 1, 2, 1, 2, 2, 2, 2, 1, 3, 2, 2, 1, 2, 2, 2, 0, 1, 2, 1, 3, 2, 2, 2, 2, 3, 2, 3, 2, 2, 3, 2, 1, 4, 3, 2, 2, 3, 2, 3, 1, 2, 3, 3, 2, 3, 2, 3, 0, 1, 2, 1, 3, 2, 2, 2, 3, 3, 2, 3, 3, 2, 3, 2, 2, 4, 3, 3, 2, 3, 3

Offset: 0

Peter Kagey, Aug 25 2016

nonn

"Coins in a Row" is a game in which players alternate picking up coins of varying denominations from the end of the row in an attempt to collect as many points as possible.

			Let [R,L,L,L] represent a game in which the first player takes the right coin, the second player takes the left coin, the first player takes the left coin, and the second player takes the left (only remaining) coin.
A066099_Row(0)    = [0];         a(0)    = 0 via [L]
A066099_Row(1)    = [1];         a(1)    = 0 via [L]
A066099_Row(3)    = [1,1];       a(3)    = 1 via [R,L]
A066099_Row(22)   = [2,1,2];     a(22)   = 2 via [L,R,L]
A066099_Row(88)   = [2,1,4];     a(88)   = 2 via [R,L,L]
A066099_Row(1418) = [2,1,4,2,2]; a(1418) = 6 via [L,R,R,R,L]

Peter Winkler, Mathematical Puzzles: A Connoisseur's Collection, A K Peters/CRC Press, 2003, pages 1-2.

Peter Kagey, Table of n, a(n) for n = 0..10000

Cf. A276163, A276164, A276165, A276166.

minimaxDifference [] = 0
minimaxDifference as = max (head as - minimaxDifference (tail as)) (last as - minimaxDifference (init as))
minimaxScore2 as = (sum as - minimaxDifference as) `div` 2
a276167 = minimaxScore2 . a066099_row

a(n) = A029837(n + 1) - A276166(n).

a(n) = A276166(n) - A276165(n).

A334966 Numbers k such that the k-th composition in standard order (row k of A066099) has weakly decreasing non-adjacent parts.

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, 27, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 47, 48, 49, 51, 55, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85, 86, 87

Offset: 1

Gus Wiseman, May 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.

The complement starts: 14, 26, 28, 29, 30, 44, 46, 50, ...

			The sequence 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)        39: (3,1,1,1)
   3: (1,1)        20: (2,3)          40: (2,4)
   4: (3)          21: (2,2,1)        41: (2,3,1)
   5: (2,1)        22: (2,1,2)        42: (2,2,2)
   6: (1,2)        23: (2,1,1,1)      43: (2,2,1,1)
   7: (1,1,1)      24: (1,4)          45: (2,1,2,1)
   8: (4)          25: (1,3,1)        47: (2,1,1,1,1)
   9: (3,1)        27: (1,2,1,1)      48: (1,5)
  10: (2,2)        31: (1,1,1,1,1)    49: (1,4,1)
  11: (2,1,1)      32: (6)            51: (1,3,1,1)
  12: (1,3)        33: (5,1)          55: (1,2,1,1,1)
  13: (1,2,1)      34: (4,2)          63: (1,1,1,1,1,1)
  15: (1,1,1,1)    35: (4,1,1)        64: (7)
  16: (5)          36: (3,3)          65: (6,1)
For example, (2,3,1,2) is such a composition because the non-adjacent pairs are (2,1), (2,2), (3,2), all of which are weakly decreasing, so 166 is in the sequence

The case of normal sequences appears to be A028859.
Strict compositions are A032020.
A version for ordered set partitions is A332872.
These compositions are enumerated by A333148.
The strict case is enumerated by A333150.
Cf. A072706, A072707, A227038, A332834, A333193.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
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

cp's OEIS Frontend

A375295 Numbers k such that the leaders of maximal weakly increasing runs in the k-th composition in standard order (row k of A066099) are not strictly decreasing.

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

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A335374 Numbers k such that the k-th composition in standard order (A066099) is not co-unimodal.

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

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A276165 a(n) is the first-player score difference of a "Coins in a Row" game over the n-th row of A066099 using a minimax strategy.

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A276166 a(n) is the first player's score in a "Coins in a Row" game over the n-th row of A066099 using a minimax strategy.

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A276167 a(n) is the second player's score in a "Coins in a Row" game over the n-th row of A066099 using a minimax strategy.

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