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

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

Author

Gus Wiseman, Jun 17 2020

Keywords

• nonn

Comments

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.

Examples

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

Links

• 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.

Crossrefs

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.

Programs

Mathematica

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

Formula

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

Extensions

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