OFFSET
0,3
COMMENTS
Also the number of (1,1,2)-avoiding or (2,1,1)-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..1000
Wikipedia, Permutation pattern
FORMULA
a(n > 0) = 2^(n - 1) - A335470(n).
a(n) = F(n,n,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 > 0 with F(0,m,k)=1 and F(n,0,k)=0 otherwise. - Andrew Howroyd, Dec 31 2020
EXAMPLE
The a(0) = 1 through a(5) = 13 compositions:
() (1) (2) (3) (4) (5)
(11) (12) (13) (14)
(21) (22) (23)
(111) (31) (32)
(112) (41)
(211) (113)
(1111) (122)
(212)
(221)
(311)
(1112)
(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, m=n); if(m==0, 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, n, 1)} \\ Andrew Howroyd, Dec 31 2020
CROSSREFS
The version for patterns is A001710.
The version for prime indices is A335449.
These compositions are ranked by A335467.
The complement A335470 is the matching version.
The (2,1,2)-avoiding version is A335473.
Compositions are counted by A011782.
Compositions avoiding (1,2,3) are counted by A102726.
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.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 17 2020
EXTENSIONS
Terms a(21) and beyond from Andrew Howroyd, Dec 31 2020
STATUS
approved