A335479 -id:A335479 - cp's OEIS frontend

A335481 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (2,1,3).

44, 88, 89, 92, 108, 152, 172, 176, 177, 178, 179, 180, 184, 185, 188, 216, 217, 220, 236, 296, 300, 304, 305, 312, 332, 344, 345, 348, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 364, 368, 369, 370, 371, 372, 376, 377, 380, 408, 428, 432, 433, 434, 435

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:
   44: (2,1,3)
   88: (2,1,4)
   89: (2,1,3,1)
   92: (2,1,1,3)
  108: (1,2,1,3)
  152: (3,1,4)
  172: (2,2,1,3)
  176: (2,1,5)
  177: (2,1,4,1)
  178: (2,1,3,2)
  179: (2,1,3,1,1)
  180: (2,1,2,3)
  184: (2,1,1,4)
  185: (2,1,1,3,1)
  188: (2,1,1,1,3)

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 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_,_}/;y

A335484 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (3,2,1).

Original entry on oeis.org

37, 69, 75, 77, 101, 133, 137, 139, 141, 149, 150, 151, 155, 157, 165, 197, 203, 205, 229, 261, 265, 267, 269, 274, 275, 277, 278, 279, 281, 283, 285, 293, 297, 299, 300, 301, 302, 303, 309, 310, 311, 315, 317, 325, 331, 333, 357, 389, 393, 395, 397, 405, 406

Offset: 1

Gus Wiseman, Jun 18 2020

nonn

			The sequence of terms together with the corresponding compositions begins:
   37: (3,2,1)
   69: (4,2,1)
   75: (3,2,1,1)
   77: (3,1,2,1)
  101: (1,3,2,1)
  133: (5,2,1)
  137: (4,3,1)
  139: (4,2,1,1)
  141: (4,1,2,1)
  149: (3,2,2,1)
  150: (3,2,1,2)
  151: (3,2,1,1,1)
  155: (3,1,2,1,1)
  157: (3,1,1,2,1)
  165: (2,3,2,1)

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

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

A375406 Number of integer compositions of n that match the dashed pattern 3-12.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 4, 14, 41, 110, 278, 673, 1576, 3599, 8055, 17732, 38509, 82683, 175830, 370856, 776723, 1616945, 3348500, 6902905, 14174198, 29004911, 59175625, 120414435, 244468774, 495340191, 1001911626, 2023473267, 4081241473, 8222198324, 16548146045, 33276169507

Offset: 0

Gus Wiseman, Aug 22 2024

nonn

First differs from the non-dashed version A335514 at a(9) = 41, A335514(9) = 42, due to the composition (3,1,3,2).

Also the number of integer compositions of n whose leaders of weakly decreasing runs are not weakly increasing. For example, the composition q = (1,1,2,1,2,2,1,3) has maximal weakly decreasing runs ((1,1),(2,1),(2,2,1),(3)), with leaders (1,2,2,3), which are weakly increasing, so q is not counted under a(13); also q does not match 3-12. On the other hand, the reverse is (3,1,2,2,1,2,1,1), with maximal weakly decreasing runs ((3,1),(2,2,1),(2,1,1)), with leaders (3,2,2), which are not weakly increasing, so it is counted under a(13); meanwhile it matches 3-12, as required.

			The a(0) = 0 through a(8) = 14 compositions:
  .  .  .  .  .  .  (312)  (412)   (413)
                           (1312)  (512)
                           (3112)  (1412)
                           (3121)  (2312)
                                   (3122)
                                   (3212)
                                   (4112)
                                   (4121)
                                   (11312)
                                   (13112)
                                   (13121)
                                   (31112)
                                   (31121)
                                   (31211)

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 A056823.
The complement is counted by A188900.
The non-dashed version is A335514, ranks A335479.
Ranks are positions of non-weakly increasing rows in A374740.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
Counting compositions by number of runs: A238130, A238279, A333755.
A373949 counts compositions by run-compressed sum, opposite A373951.
Cf. A106356, A188920, A189076, A189077, A238343, A333213, A335548, A374629, A374637, A374679, A374748.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !LessEqual@@First/@Split[#,GreaterEqual]&]],{n,0,15}]
- or -
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], MatchQ[#,{_,z_,_,x_,y_,_}/;x

a(n>0) = 2^(n-1) - A188900(n).

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A335481 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (2,1,3).

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A335484 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (3,2,1).

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A375406 Number of integer compositions of n that match the dashed pattern 3-12.

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