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

Author

Gus Wiseman, Jun 18 2020

Keywords

• nonn

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).

Examples

			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)
		

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.

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.

Programs

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
				

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

Formula

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

Extensions

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