A333218 -id:A333218 - cp's OEIS frontend

A335514 Number of (1,2,3)-matching compositions of n.

0, 0, 0, 0, 0, 0, 1, 4, 14, 42, 114, 292, 714, 1686, 3871, 8696, 19178, 41667, 89386, 189739, 399144, 833290, 1728374, 3565148, 7319212, 14965880, 30496302, 61961380, 125577752, 253971555, 512716564, 1033496947, 2080572090, 4183940550, 8406047907, 16875834728

Offset: 0

Gus Wiseman, Jun 22 2020

nonn

			The a(6) = 1 through a(8) = 14 compositions:
  (1,2,3)  (1,2,4)    (1,2,5)
           (1,1,2,3)  (1,3,4)
           (1,2,1,3)  (1,1,2,4)
           (1,2,3,1)  (1,2,1,4)
                      (1,2,2,3)
                      (1,2,3,2)
                      (1,2,4,1)
                      (2,1,2,3)
                      (1,1,1,2,3)
                      (1,1,2,1,3)
                      (1,1,2,3,1)
                      (1,2,1,1,3)
                      (1,2,1,3,1)
                      (1,2,3,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..500
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The version for permutations is A056986.
The avoiding version is A102726.
These compositions are ranked by A335479.
The version for patterns is A335515.
The version for prime indices is A335520.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Patterns matched by compositions are counted by A335456.
Cf. A011782, A032020, A106356, A226316, A269134, A333755, A335465, A335521.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],MatchQ[#,{_,x_,_,y_,_,z_,_}/;x

a(n > 0) = 2^(n - 1) - A102726(n).

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2020

A087117 Number of zeros in the longest string of consecutive zeros in the binary representation of n.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Aug 14 2003

The following four statements are equivalent: a(n) = 0; n = 2^k - 1 for some k > 0; A087116(n) = 0; A023416(n) = 0.

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. Then a(k) is the maximum part of this composition, minus one. The maximum part is A333766(k). - Gus Wiseman, Apr 09 2020

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to binary expansion of n

Cf. A023416, A007088, A007814.
Cf. A025480, A038374, A090046, A090047, A090048, A090049, A090050.
Positions of zeros are A000225.
Positions of terms <= 1 are A003754.
Positions of terms > 0 are A062289.
Positions of first appearances are A131577.
The version for prime indices is A252735.
The proper maximum is A333766.
The version for minimum is A333767.
Maximum prime index is A061395.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Runs are counted by A124767.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Runs-resistance is A333628.
- Weakly decreasing compositions are A114994.
- Weakly increasing compositions are A225620.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
Cf. A029931, A048793, A061395, A087117, A228351, A328594, A333217, A333218, A333219, A333767, A333768.

import Data.List (unfoldr, group)
a087117 0       = 1
a087117 n
  | null $ zs n = 0
  | otherwise   = maximum $ map length $ zs n where
  zs = filter ((== 0) . head) . group .
       unfoldr (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2)
-- Reinhard Zumkeller, May 01 2012

A087117 := proc(n)
    local d,l,zlen ;
    if n = 0 then
        return 1 ;
    end if;
    d := convert(n,base,2) ;
    for l from nops(d)-1 to 0 by -1 do
        zlen := [seq(0,i=1..l)] ;
        if verify(zlen,d,'sublist') then
            return l ;
        end if;
    end do:
    return 0 ;
end proc; # R. J. Mathar, Nov 05 2012

nz[n_]:=Max[Length/@Select[Split[IntegerDigits[n,2]],MemberQ[#,0]&]]; Array[nz,110,0]/.-\[Infinity]->0 (* Harvey P. Dale, Sep 05 2017 *)

h(n)=if(n<2, return(0)); my(k=valuation(n,2)); if(k, max(h(n>>k), k), n++; n>>=valuation(n,2); h(n-1))
a(n)=if(n,h(n),1) \\ Charles R Greathouse IV, Apr 06 2022

a(n) = max(A007814(n), a(A025480(n-1))) for n >= 2. - Robert Israel, Feb 19 2017
a(2n+1) = a(n) (n>=1); indeed, the binary form of 2n+1 consists of the binary form of n with an additional 1 at the end - Emeric Deutsch, Aug 18 2017
For n > 0, a(n) = A333766(n) - 1. - Gus Wiseman, Apr 09 2020

A333768 Minimum part of the n-th composition in standard order. a(0) = 0.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Apr 06 2020

nonn

One plus the shortest run of 0's after a 1 in the binary expansion of n > 0.

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 148th composition in standard order is (3,2,3), so a(148) = 2.

Positions of first appearances (ignoring index 0) are A000079.
Positions of terms > 1 are A022340.
The version for prime indices is A055396.
The  maximum part is given by A333766.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Compositions without 1's are A022340.
- Sum is A070939.
- Product is A124758.
- Runs are counted by A124767.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Runs-resistance is A333628.
- Weakly decreasing compositions are A114994.
- Weakly increasing compositions are A225620.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
Cf. A029931, A048793, A087117, A228351, A328594, A333217, A333218, A333219, A333632, A333767.

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

For n > 0, a(n) = A333767(n) + 1.

A335479 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,2,3).

Original entry on oeis.org

52, 104, 105, 108, 116, 180, 200, 208, 209, 210, 211, 212, 216, 217, 220, 232, 233, 236, 244, 308, 328, 360, 361, 364, 372, 400, 401, 404, 408, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 428, 432, 433, 434, 435, 436, 440, 441, 444, 456, 464, 465, 466

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).

			The sequence of terms together with the corresponding compositions begins:
   52: (1,2,3)
  104: (1,2,4)
  105: (1,2,3,1)
  108: (1,2,1,3)
  116: (1,1,2,3)
  180: (2,1,2,3)
  200: (1,3,4)
  208: (1,2,5)
  209: (1,2,4,1)
  210: (1,2,3,2)
  211: (1,2,3,1,1)
  212: (1,2,2,3)
  216: (1,2,1,4)
  217: (1,2,1,3,1)
  220: (1,2,1,1,3)

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 version counting permutations is A056986.
Patterns matching this pattern are counted by A335515 (by length).
Permutations of prime indices matching this pattern are counted by A335520.
These compositions are counted by A335514 (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.
Other permutations:
- A335479 (1,2,3)
- A335480 (1,3,2)
- A335481 (2,1,3)
- A335482 (2,3,1)
- A335483 (3,1,2)
- A335484 (3,2,1)
Cf. A034691, A056986, A108917, A114994, A158005, 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_,_,y_,_,z_,_}/;x

A335470 Number of compositions of n matching the pattern (1,2,1).

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 9, 24, 61, 141, 322, 713, 1543, 3289, 6907, 14353, 29604, 60640, 123522, 250645, 506808, 1022197, 2057594, 4135358, 8301139, 16648165, 33364948, 66831721, 133814251, 267850803, 536026676, 1072528081, 2145745276, 4292485526, 8586405894, 17174865820

Offset: 0

Gus Wiseman, Jun 17 2020

nonn

Also the number of (1,1,2)-matching or (2,1,1)-matching 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).
A composition of n is a finite sequence of positive integers summing to n.

			The a(4) = 1 through a(6) = 9 compositions:
  (121)  (131)   (141)
         (1121)  (1131)
         (1211)  (1212)
                 (1221)
                 (1311)
                 (2121)
                 (11121)
                 (11211)
                 (12111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The version for prime indices is A335446.
These compositions are ranked by A335466.
The complement A335471 is the avoiding version.
The (2,1,2)-matching version is A335472.
The version for patterns is A335509.
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.
Compositions are counted by A011782.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by compositions are counted by A335456.
Minimal patterns avoided by a standard composition are counted by A335465.
Compositions matching (1,2,3) are counted by A335514.
Cf. A261982, A034691, A056986, A106356, A238279, A333755, A335454.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],MatchQ[#,{_,x_,_,y_,_,x_,_}/;x

a(n > 0) = 2^(n - 1) - A335471(n).

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2020

A335471 Number of compositions of n avoiding the pattern (1,2,1).

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 23, 40, 67, 115, 190, 311, 505, 807, 1285, 2031, 3164, 4896, 7550, 11499, 17480, 26379, 39558, 58946, 87469, 129051, 189484, 277143, 403477, 584653, 844236, 1213743, 1738372, 2481770, 3528698, 5003364, 7070225, 9958387, 13982822, 19580613, 27333403

Offset: 0

Gus Wiseman, Jun 17 2020

nonn

Also the number of (1,1,2)-avoiding or (2,1,1)-avoiding 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).
A composition of n is a finite sequence of positive integers summing to n.

			The a(0) = 1 through a(5) = 13 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (211)   (113)
                        (1111)  (122)
                                (212)
                                (221)
                                (311)
                                (1112)
                                (2111)
                                (11111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The version for patterns is A001710.
The version for prime indices is A335449.
These compositions are ranked by A335467.
The complement A335470 is the matching version.
The (2,1,2)-avoiding version is A335473.
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.
Compositions are counted by A011782.
Compositions avoiding (1,2,3) are counted by A102726.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by compositions are counted by A335456.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A106356, A232464, A238279, A333213, A333755.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,_,y_,_,x_,_}/;x

a(n)={local(Cache=Map()); my(F(n,m,k)=if(m>n, m=n); if(m==0, n==0, my(hk=[n,m,k], z); if(!mapisdefined(Cache,hk,&z), z=self()(n,m-1,k) + k*sum(i=1,n\m, self()(n-i*m, m-1, k+i)); mapput(Cache, hk, z)); z)); F(n,n,1)} \\ Andrew Howroyd, Dec 31 2020

a(n > 0) = 2^(n - 1) - A335470(n).

a(n) = F(n,n,1) where F(n,m,k) = F(n,m-1,k) + k*(Sum_{i=1..floor(n/m)} F(n-i*m, m-1, k+i)) for m > 0 with F(0,m,k)=1 and F(n,0,k)=0 otherwise. - Andrew Howroyd, Dec 31 2020

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2020

A335473 Number of compositions of n avoiding the pattern (2,1,2).

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 29, 55, 103, 190, 347, 630, 1134, 2028, 3585, 6291, 10950, 18944, 32574, 55692, 94618, 159758, 268147, 447502, 743097, 1227910, 2020110, 3308302, 5394617, 8757108, 14155386, 22784542, 36529813, 58343498, 92850871, 147254007, 232750871, 366671436

Offset: 0

Gus Wiseman, Jun 17 2020

nonn

Also the number of (1,2,2) or (2,2,1)-avoiding 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).
A composition of n is a finite sequence of positive integers summing to n.

			The a(0) = 1 through a(5) = 15 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (211)   (122)
                        (1111)  (131)
                                (221)
                                (311)
                                (1112)
                                (1121)
                                (1211)
                                (2111)
                                (11111)

Andrew Howroyd, Table of n, a(n) for n = 0..200
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The version for patterns is A001710.
The version for prime indices is A335450.
These compositions are ranked by A335469.
The (1,2,1)-avoiding version is A335471.
The complement A335472 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.
Compositions are counted by A011782.
Compositions avoiding (1,2,3) are counted by A102726.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by compositions are counted by A335456.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A261982, A034691, A056986, A106356, A232464, A238279, A333755.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,_,y_,_,x_,_}/;x>y]&]],{n,0,10}]

a(n)={local(Cache=Map()); my(F(n,m,k) = if(m>n, n==0, my(hk=[n,m,k], z); if(!mapisdefined(Cache,hk,&z), z=self()(n,m+1,k) + k*sum(i=1,n\m, self()(n-i*m, m+1, k+i)); mapput(Cache, hk, z)); z)); F(n,1,1)} \\ Andrew Howroyd, Dec 31 2020

a(n > 0) = 2^(n - 1) - A335472(n).

a(n) = F(n,1,1) where F(n,m,k) = F(n,m+1,k) + k*(Sum_{i=1..floor(n/m)} F(n-i*m, m+1, k+i)) for m <= n with F(0,m,k)=1 and F(n,m,k)=0 otherwise. - Andrew Howroyd, Dec 31 2020

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2020

A335480 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,3,2).

Original entry on oeis.org

50, 98, 101, 102, 114, 178, 194, 196, 197, 198, 202, 203, 205, 206, 210, 226, 229, 230, 242, 306, 324, 354, 357, 358, 370, 386, 388, 389, 390, 393, 394, 395, 396, 397, 398, 402, 404, 405, 406, 407, 410, 411, 413, 414, 418, 421, 422, 434, 450, 452, 453, 454

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).

			The sequence of terms together with the corresponding compositions begins:
   50: (1,3,2)
   98: (1,4,2)
  101: (1,3,2,1)
  102: (1,3,1,2)
  114: (1,1,3,2)
  178: (2,1,3,2)
  194: (1,5,2)
  196: (1,4,3)
  197: (1,4,2,1)
  198: (1,4,1,2)
  202: (1,3,2,2)
  203: (1,3,2,1,1)
  205: (1,3,1,2,1)
  206: (1,3,1,1,2)
  210: (1,2,3,2)

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 counting permutations is A056986.
Patterns matching this pattern are counted by A335515 (by length).
Permutations of prime indices matching this pattern are counted by A335520.
These compositions are counted by A335514 (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.
Permutations matching (1,3,2,4) are counted by A158009.
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.
Other permutations:
- A335479 (1,2,3)
- A335480 (1,3,2)
- A335481 (2,1,3)
- A335482 (2,3,1)
- A335483 (3,1,2)
- A335484 (3,2,1)
Cf. A034691, A056986, A108917, A114994, A158005, 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_,_,y_,_,z_,_}/;x

A335482 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (2,3,1).

Original entry on oeis.org

41, 81, 83, 89, 105, 145, 161, 163, 165, 166, 167, 169, 177, 179, 185, 209, 211, 217, 233, 289, 290, 291, 297, 305, 321, 323, 325, 326, 327, 329, 331, 332, 333, 334, 335, 337, 339, 345, 353, 355, 357, 358, 359, 361, 369, 371, 377, 401, 417, 419, 421, 422, 423

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).

			The sequence of terms together with the corresponding compositions begins:
   41: (2,3,1)
   81: (2,4,1)
   83: (2,3,1,1)
   89: (2,1,3,1)
  105: (1,2,3,1)
  145: (3,4,1)
  161: (2,5,1)
  163: (2,4,1,1)
  165: (2,3,2,1)
  166: (2,3,1,2)
  167: (2,3,1,1,1)
  169: (2,2,3,1)
  177: (2,1,4,1)
  179: (2,1,3,1,1)
  185: (2,1,1,3,1)

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 version counting permutations is A056986.
Patterns matching this pattern are counted by A335515 (by length).
Permutations of prime indices matching this pattern are counted by A335520.
These compositions are counted by A335514 (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.
Permutations matching (1,3,2,4) are counted by A158009.
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.
Other permutations:
- A335479 (1,2,3)
- A335480 (1,3,2)
- A335481 (2,1,3)
- A335482 (2,3,1)
- A335483 (3,1,2)
- A335484 (3,2,1)
Cf. A034691, A056986, A108917, A114994, A158005, 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_,_,y_,_,z_,_}/;z

A335485 Numbers k such that the k-th composition in standard order (A066099) is not weakly decreasing.

Original entry on oeis.org

6, 12, 13, 14, 20, 22, 24, 25, 26, 27, 28, 29, 30, 38, 40, 41, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 70, 72, 76, 77, 78, 80, 81, 82, 83, 84, 86, 88, 89, 90, 91, 92, 93, 94, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106

Offset: 1

Gus Wiseman, Jun 18 2020

nonn

Also compositions matching the pattern (1,2).

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

			The sequence of terms together with the corresponding compositions begins:
   6: (1,2)
  12: (1,3)
  13: (1,2,1)
  14: (1,1,2)
  20: (2,3)
  22: (2,1,2)
  24: (1,4)
  25: (1,3,1)
  26: (1,2,2)
  27: (1,2,1,1)
  28: (1,1,3)
  29: (1,1,2,1)
  30: (1,1,1,2)
  38: (3,1,2)
  40: (2,4)

Keiichi Shigechi, Noncommutative crossing partitions, arXiv:2211.10958 [math.CO], 2022.
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 A114994 is the avoiding version.
The (2,1)-matching version is A335486.
Patterns matching this pattern are counted by A002051 (by length).
Permutations of prime indices matching this pattern are counted by A335447.
These compositions are counted by A056823 (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, A335456, A335458.

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

cp's OEIS Frontend

A335514 Number of (1,2,3)-matching compositions of n.

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A087117 Number of zeros in the longest string of consecutive zeros in the binary representation of n.

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A333768 Minimum part of the n-th composition in standard order. a(0) = 0.

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A335479 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,2,3).

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A335470 Number of compositions of n matching the pattern (1,2,1).

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A335471 Number of compositions of n avoiding the pattern (1,2,1).

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A335473 Number of compositions of n avoiding the pattern (2,1,2).

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