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

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

1,1

Comments

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).

Links

Table of n, a(n) for n=1..53.

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.

Example

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)

Mathematica

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

Crossrefs

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.

Sequence in context: A352346 A214476 A121738 * A043331 A023701 A031459

Adjacent sequences: A335472 A335473 A335474 * A335476 A335477 A335478

Keyword

nonn

Author

Gus Wiseman, Jun 18 2020

Status

approved