A335470 - OEIS

%I #14 Dec 31 2020 15:36:11

%S 0,0,0,0,1,3,9,24,61,141,322,713,1543,3289,6907,14353,29604,60640,

%T 123522,250645,506808,1022197,2057594,4135358,8301139,16648165,

%U 33364948,66831721,133814251,267850803,536026676,1072528081,2145745276,4292485526,8586405894,17174865820

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

%C Also the number of (1,1,2)-matching or (2,1,1)-matching compositions.

%C 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).

%C A composition of n is a finite sequence of positive integers summing to n.

%H Andrew Howroyd, <a href="/A335470/b335470.txt">Table of n, a(n) for n = 0..1000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>

%H Gus Wiseman, <a href="https://oeis.org/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>

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

%e The a(4) = 1 through a(6) = 9 compositions:

%e   (121)  (131)   (141)

%e          (1121)  (1131)

%e          (1211)  (1212)

%e                  (1221)

%e                  (1311)

%e                  (2121)

%e                  (11121)

%e                  (11211)

%e                  (12111)

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

%Y The version for prime indices is A335446.

%Y These compositions are ranked by A335466.

%Y The complement A335471 is the avoiding version.

%Y The (2,1,2)-matching version is A335472.

%Y The version for patterns is A335509.

%Y Constant patterns are counted by A000005 and ranked by A272919.

%Y Permutations are counted by A000142 and ranked by A333218.

%Y Patterns are counted by A000670 and ranked by A333217.

%Y Compositions are counted by A011782.

%Y Non-unimodal compositions are counted by A115981 and ranked by A335373.

%Y Combinatory separations are counted by A269134.

%Y Patterns matched by compositions are counted by A335456.

%Y Minimal patterns avoided by a standard composition are counted by A335465.

%Y Compositions matching (1,2,3) are counted by A335514.

%Y Cf. A261982, A034691, A056986, A106356, A238279, A333755, A335454.

%K nonn

%O 0,6

%A _Gus Wiseman_, Jun 17 2020

%E Terms a(21) and beyond from _Andrew Howroyd_, Dec 31 2020