OFFSET
0,6
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.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1000
Wikipedia, Permutation pattern
FORMULA
a(n > 0) = 2^(n - 1) - A335471(n).
EXAMPLE
The a(4) = 1 through a(6) = 9 compositions:
(121) (131) (141)
(1121) (1131)
(1211) (1212)
(1221)
(1311)
(2121)
(11121)
(11211)
(12111)
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], MatchQ[#, {___, x_, ___, y_, ___, x_, ___}/; x<y]&]], {n, 0, 10}]
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.
Compositions are counted by A011782.
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.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 17 2020
EXTENSIONS
Terms a(21) and beyond from Andrew Howroyd, Dec 31 2020
STATUS
approved