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%I #13 Dec 31 2020 15:36:30
%S 0,0,0,0,0,1,3,9,25,66,165,394,914,2068,4607,10093,21818,46592,98498,
%T 206452,429670,888818,1829005,3746802,7645511,15549306,31534322,
%U 63800562,128823111,259678348,522715526,1050957282,2110953835,4236623798,8497083721,17032615177
%N Number of compositions of n matching the pattern (2,1,2).
%C Also the number of (1,2,2) or (2,2,1)-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="/A335472/b335472.txt">Table of n, a(n) for n = 0..200</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) - A335473(n).
%e The a(5) = 1 through a(7) = 9 compositions:
%e (212) (1212) (313)
%e (2112) (2122)
%e (2121) (2212)
%e (11212)
%e (12112)
%e (12121)
%e (21112)
%e (21121)
%e (21211)
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],MatchQ[#,{___,x_,___,y_,___,x_,___}/;x>y]&]],{n,0,10}]
%Y The version for prime indices is A335453.
%Y These compositions are ranked by A335468.
%Y The (1,2,1)-matching version is A335470.
%Y The complement A335473 is the avoiding version.
%Y The version for patterns is A335509.
%Y Constant patterns are counted by A000005 and ranked by A272919.
%Y Patterns are counted by A000670 and ranked by A333217.
%Y Permutations are counted by A000142 and ranked by A333218.
%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.
%K nonn
%O 0,7
%A _Gus Wiseman_, Jun 17 2020
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