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

%I A335470 #14 Dec 31 2020 15:36:11
%S A335470 0,0,0,0,1,3,9,24,61,141,322,713,1543,3289,6907,14353,29604,60640,
%T A335470 123522,250645,506808,1022197,2057594,4135358,8301139,16648165,
%U A335470 33364948,66831721,133814251,267850803,536026676,1072528081,2145745276,4292485526,8586405894,17174865820
%N A335470 Number of compositions of n matching the pattern (1,2,1).
%C A335470 Also the number of (1,1,2)-matching or (2,1,1)-matching compositions.
%C A335470 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 A335470 A composition of n is a finite sequence of positive integers summing to n.
%H A335470 Andrew Howroyd, <a href="/A335470/b335470.txt">Table of n, a(n) for n = 0..1000</a>
%H A335470 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>
%H A335470 Gus Wiseman, <a href="https://oeis.org/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>
%F A335470 a(n > 0) = 2^(n - 1) - A335471(n).
%e A335470 The a(4) = 1 through a(6) = 9 compositions:
%e A335470   (121)  (131)   (141)
%e A335470          (1121)  (1131)
%e A335470          (1211)  (1212)
%e A335470                  (1221)
%e A335470                  (1311)
%e A335470                  (2121)
%e A335470                  (11121)
%e A335470                  (11211)
%e A335470                  (12111)
%t A335470 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],MatchQ[#,{___,x_,___,y_,___,x_,___}/;x<y]&]],{n,0,10}]
%Y A335470 The version for prime indices is A335446.
%Y A335470 These compositions are ranked by A335466.
%Y A335470 The complement A335471 is the avoiding version.
%Y A335470 The (2,1,2)-matching version is A335472.
%Y A335470 The version for patterns is A335509.
%Y A335470 Constant patterns are counted by A000005 and ranked by A272919.
%Y A335470 Permutations are counted by A000142 and ranked by A333218.
%Y A335470 Patterns are counted by A000670 and ranked by A333217.
%Y A335470 Compositions are counted by A011782.
%Y A335470 Non-unimodal compositions are counted by A115981 and ranked by A335373.
%Y A335470 Combinatory separations are counted by A269134.
%Y A335470 Patterns matched by compositions are counted by A335456.
%Y A335470 Minimal patterns avoided by a standard composition are counted by A335465.
%Y A335470 Compositions matching (1,2,3) are counted by A335514.
%Y A335470 Cf. A261982, A034691, A056986, A106356, A238279, A333755, A335454.
%K A335470 nonn
%O A335470 0,6
%A A335470 _Gus Wiseman_, Jun 17 2020
%E A335470 Terms a(21) and beyond from _Andrew Howroyd_, Dec 31 2020