

A335473


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


9



1, 1, 2, 4, 8, 15, 29, 55, 103, 190, 347, 630, 1134, 2028, 3585, 6291, 10950, 18944, 32574, 55692, 94618, 159758, 268147, 447502, 743097, 1227910, 2020110, 3308302, 5394617, 8757108, 14155386, 22784542, 36529813, 58343498, 92850871, 147254007, 232750871, 366671436
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OFFSET

0,3


COMMENTS

Also the number of (1,2,2) or (2,2,1)avoiding 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

Andrew Howroyd, Table of n, a(n) for n = 0..200
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.


FORMULA

a(n > 0) = 2^(n  1)  A335472(n).
a(n) = F(n,1,1) where F(n,m,k) = F(n,m+1,k) + k*(Sum_{i=1..floor(n/m)} F(ni*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


EXAMPLE

The a(0) = 1 through a(5) = 15 compositions:
() (1) (2) (3) (4) (5)
(11) (12) (13) (14)
(21) (22) (23)
(111) (31) (32)
(112) (41)
(121) (113)
(211) (122)
(1111) (131)
(221)
(311)
(1112)
(1121)
(1211)
(2111)
(11111)


MATHEMATICA

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


PROG

(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()(ni*m, m+1, k+i)); mapput(Cache, hk, z)); z)); F(n, 1, 1)} \\ Andrew Howroyd, Dec 31 2020


CROSSREFS

The version for patterns is A001710.
The version for prime indices is A335450.
These compositions are ranked by A335469.
The (1,2,1)avoiding version is A335471.
The complement A335472 is the matching version.
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Compositions are counted by A011782.
Compositions avoiding (1,2,3) are counted by A102726.
Nonunimodal 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.
Cf. A261982, A034691, A056986, A106356, A232464, A238279, A333755.
Sequence in context: A217733 A208976 A278554 * A224959 A108564 A066369
Adjacent sequences: A335470 A335471 A335472 * A335474 A335475 A335476


KEYWORD

nonn


AUTHOR

Gus Wiseman, Jun 17 2020


EXTENSIONS

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2020


STATUS

approved



