A335514 Number of (1,2,3)-matching compositions of n.

0, 0, 0, 0, 0, 0, 1, 4, 14, 42, 114, 292, 714, 1686, 3871, 8696, 19178, 41667, 89386, 189739, 399144, 833290, 1728374, 3565148, 7319212, 14965880, 30496302, 61961380, 125577752, 253971555, 512716564, 1033496947, 2080572090, 4183940550, 8406047907, 16875834728

Offset

0,8

Links

Andrew Howroyd, Table of n, a(n) for n = 0..500

Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Formula

a(n > 0) = 2^(n - 1) - A102726(n).

Example

The a(6) = 1 through a(8) = 14 compositions:
  (1,2,3)  (1,2,4)    (1,2,5)
           (1,1,2,3)  (1,3,4)
           (1,2,1,3)  (1,1,2,4)
           (1,2,3,1)  (1,2,1,4)
                      (1,2,2,3)
                      (1,2,3,2)
                      (1,2,4,1)
                      (2,1,2,3)
                      (1,1,1,2,3)
                      (1,1,2,1,3)
                      (1,1,2,3,1)
                      (1,2,1,1,3)
                      (1,2,1,3,1)
                      (1,2,3,1,1)

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], MatchQ[#, {___, x_, ___, y_, ___, z_, ___}/; x<y<z]&]], {n, 0, 10}]

Crossrefs

The version for permutations is A056986.

The avoiding version is A102726.

These compositions are ranked by A335479.

The version for patterns is A335515.

The version for prime indices is A335520.

Permutations are counted by A000142 and ranked by A333218.

Patterns are counted by A000670 and ranked by A333217.

Patterns matched by compositions are counted by A335456.

Cf. A011782, A032020, A106356, A226316, A269134, A333755, A335465, A335521.

Sequence in context: A295201 A309296 A124616 * A221058 A347583 A124617

Adjacent sequences: A335511 A335512 A335513 * A335515 A335516 A335517

Keyword

nonn

Author

Gus Wiseman, Jun 22 2020

Extensions

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

Status

approved