A335466 Numbers k such that the k-th composition in standard order (A066099) matches (1,2,1).

13, 25, 27, 29, 45, 49, 51, 53, 54, 55, 57, 59, 61, 77, 82, 89, 91, 93, 97, 99, 101, 102, 103, 105, 107, 108, 109, 110, 111, 113, 115, 117, 118, 119, 121, 123, 125, 141, 153, 155, 157, 162, 165, 166, 173, 177, 178, 179, 181, 182, 183, 185, 187, 189, 193, 195

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..56.

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

Example

The sequence of terms together with the 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)
  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)

Mathematica

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

Crossrefs

The complement A335467 is the avoiding version.

The (2,1,2)-matching version is A335468.

These compositions are counted by A335470.

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 and ranked by A334030.

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, A335509.

Sequence in context: A335374 A375296 A375295 * A186403 A032478 A280925

Adjacent sequences: A335463 A335464 A335465 * A335467 A335468 A335469

Keyword

nonn

Author

Gus Wiseman, Jun 15 2020

Status

approved