A335509 Number of patterns of length n matching the pattern (1,1,2).

0, 0, 0, 1, 15, 181, 2163, 27133, 364395, 5272861, 82289163, 1383131773, 24978057195, 483269202781, 9987505786443, 219821796033853, 5137810967933355, 127169580176271901, 3324712113052429323, 91585136315240091133, 2652142325158529483115, 80562824634615270041821

Offset

0,5

Comments

Also the number of (1,2,1)-matching patterns of length n.

Also the number of (2,1,2)-matching patterns of length n.

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

Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Formula

E.g.f.: 1/(2-exp(x)) - (2-2*x+x^2)/(2*(1-x)^2). - Andrew Howroyd, Dec 31 2020

Example

The a(3) = 1 through a(4) = 15 patterns:
  (1,1,2)  (1,1,1,2)
           (1,1,2,1)
           (1,1,2,2)
           (1,1,2,3)
           (1,1,3,2)
           (1,2,1,2)
           (1,2,1,3)
           (1,2,2,3)
           (1,3,1,2)
           (2,1,1,2)
           (2,1,1,3)
           (2,1,2,3)
           (2,2,1,3)
           (2,2,3,1)
           (3,1,1,2)

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_, ___, y_, ___}/; x<y]&]], {n, 0, 6}]

Prog

(PARI) seq(n)={Vec(serlaplace(1/(2-exp(x + O(x*x^n))) - (2-2*x+x^2)/(2*(1-x)^2)), -(n+1))} \\ Andrew Howroyd, Dec 31 2020

Crossrefs

The complement A001710 is the avoiding version.

Compositions matching this pattern are counted by A335470 and ranked by A335476.

Permutations of prime indices matching this pattern are counted by A335446.

Patterns are counted by A000670 and ranked by A333217.

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, A238279, A292884, A333755, A335456, A335457.

Sequence in context: A138443 A235455 A016158 * A193701 A125450 A193709

Adjacent sequences: A335506 A335507 A335508 * A335510 A335511 A335512

Keyword

nonn

Author

Gus Wiseman, Jun 18 2020

Extensions

Terms a(10) and beyond from Andrew Howroyd, Dec 31 2020

Status

approved