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
Alois P. Heinz, Table of n, a(n) for n = 0..424
Wikipedia, Permutation pattern
FORMULA
a(n) = Sum_{k=3..n} A276922(n,k). - Alois P. Heinz, Jan 28 2024
EXAMPLE
The a(3) = 1 through a(4) = 9 patterns:
(1,1,1) (1,1,1,1)
(1,1,1,2)
(1,1,2,1)
(1,2,1,1)
(1,2,2,2)
(2,1,1,1)
(2,1,2,2)
(2,2,1,2)
(2,2,2,1)
MAPLE
b:= proc(n, k) option remember; `if`(n=0, 1, add(
b(n-i, k)*binomial(n, i), i=1..min(n, k)))
end:
a:= n-> b(n$2)-b(n, 2):
seq(a(n), n=0..21); # Alois P. Heinz, Jan 28 2024
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_, ___, x_, ___, x_, ___}]&]], {n, 0, 6}]
CROSSREFS
The complement A080599 is the avoiding version.
Permutations of prime indices matching this pattern are counted by A335510.
Patterns matching the pattern (1,1) are counted by A019472.
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.
Patterns matching (1,2,3) are counted by A335515.
Cf. A034691, A056986, A232464, A238279, A292884, A333175, A333755, A335451, A335456, A335457, A335458.
Cf. A276922.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 18 2020
EXTENSIONS
a(9)-a(21) from Alois P. Heinz, Jan 28 2024
STATUS
approved