|
|
A335470
|
|
Number of compositions of n matching the pattern (1,2,1).
|
|
10
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
FORMULA
|
|
|
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.
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.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|