OFFSET
0,5
COMMENTS
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).
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..200
Wikipedia, Permutation pattern
FORMULA
EXAMPLE
The a(3) = 1 through a(4) = 19 patterns:
(1,2,3) (1,1,2,3)
(1,2,1,3)
(1,2,2,3)
(1,2,3,1)
(1,2,3,2)
(1,2,3,3)
(1,2,3,4)
(1,2,4,3)
(1,3,2,3)
(1,3,2,4)
(1,3,4,2)
(1,4,2,3)
(2,1,2,3)
(2,1,3,4)
(2,3,1,4)
(2,3,4,1)
(3,1,2,3)
(3,1,2,4)
(4,1,2,3)
MATHEMATICA
allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n], MatchQ[#, {___, x_, ___, y_, ___, z_, ___}/; x<y<z]&]], {n, 0, 6}]
PROG
(PARI) seq(n)=Vec( serlaplace(1/(2-exp(x + O(x*x^n)))) - 1/2 - 1/(1+sqrt(1-8*x+8*x^2 + O(x*x^n))), -(n+1)) \\ Andrew Howroyd, Jan 28 2024
CROSSREFS
The complement A226316 is the avoiding version.
Permutations of prime indices matching this pattern are counted by A335520.
Patterns matching the pattern (1,1) are counted by A019472.
Permutations matching (1,2,3) are counted by A056986.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 19 2020
EXTENSIONS
a(9) onwards from Andrew Howroyd, Jan 28 2024
STATUS
approved