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

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

Author

Gus Wiseman, Jun 18 2020

Keywords

• nonn

Comments

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).

Examples

			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)
		

Links

• 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

Crossrefs

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.

Programs

Mathematica

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