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A335473 Number of compositions of n avoiding the pattern (2,1,2).

%I A335473 #13 Dec 31 2020 15:36:36
%S A335473 1,1,2,4,8,15,29,55,103,190,347,630,1134,2028,3585,6291,10950,18944,
%T A335473 32574,55692,94618,159758,268147,447502,743097,1227910,2020110,
%U A335473 3308302,5394617,8757108,14155386,22784542,36529813,58343498,92850871,147254007,232750871,366671436
%N A335473 Number of compositions of n avoiding the pattern (2,1,2).
%C A335473 Also the number of (1,2,2) or (2,2,1)-avoiding compositions.
%C A335473 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 A335473 A composition of n is a finite sequence of positive integers summing to n.
%H A335473 Andrew Howroyd, <a href="/A335473/b335473.txt">Table of n, a(n) for n = 0..200</a>
%H A335473 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>
%H A335473 Gus Wiseman, <a href="https://oeis.org/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>
%F A335473 a(n > 0) = 2^(n - 1) - A335472(n).
%F A335473 a(n) = F(n,1,1) where F(n,m,k) = F(n,m+1,k) + k*(Sum_{i=1..floor(n/m)} F(n-i*m, m+1, k+i)) for m <= n with F(0,m,k)=1 and F(n,m,k)=0 otherwise. - _Andrew Howroyd_, Dec 31 2020
%e A335473 The a(0) = 1 through a(5) = 15 compositions:
%e A335473   ()  (1)  (2)   (3)    (4)     (5)
%e A335473            (11)  (12)   (13)    (14)
%e A335473                  (21)   (22)    (23)
%e A335473                  (111)  (31)    (32)
%e A335473                         (112)   (41)
%e A335473                         (121)   (113)
%e A335473                         (211)   (122)
%e A335473                         (1111)  (131)
%e A335473                                 (221)
%e A335473                                 (311)
%e A335473                                 (1112)
%e A335473                                 (1121)
%e A335473                                 (1211)
%e A335473                                 (2111)
%e A335473                                 (11111)
%t A335473 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{___,x_,___,y_,___,x_,___}/;x>y]&]],{n,0,10}]
%o A335473 (PARI) a(n)={local(Cache=Map()); my(F(n,m,k) = if(m>n, n==0, my(hk=[n,m,k], z); if(!mapisdefined(Cache,hk,&z), z=self()(n,m+1,k) + k*sum(i=1,n\m, self()(n-i*m, m+1, k+i)); mapput(Cache, hk, z)); z)); F(n,1,1)} \\ _Andrew Howroyd_, Dec 31 2020
%Y A335473 The version for patterns is A001710.
%Y A335473 The version for prime indices is A335450.
%Y A335473 These compositions are ranked by A335469.
%Y A335473 The (1,2,1)-avoiding version is A335471.
%Y A335473 The complement A335472 is the matching version.
%Y A335473 Constant patterns are counted by A000005 and ranked by A272919.
%Y A335473 Permutations are counted by A000142 and ranked by A333218.
%Y A335473 Patterns are counted by A000670 and ranked by A333217.
%Y A335473 Compositions are counted by A011782.
%Y A335473 Compositions avoiding (1,2,3) are counted by A102726.
%Y A335473 Non-unimodal compositions are counted by A115981 and ranked by A335373.
%Y A335473 Combinatory separations are counted by A269134.
%Y A335473 Patterns matched by compositions are counted by A335456.
%Y A335473 Minimal patterns avoided by a standard composition are counted by A335465.
%Y A335473 Cf. A261982, A034691, A056986, A106356, A232464, A238279, A333755.
%K A335473 nonn
%O A335473 0,3
%A A335473 _Gus Wiseman_, Jun 17 2020
%E A335473 Terms a(21) and beyond from _Andrew Howroyd_, Dec 31 2020