A335515 Number of patterns of length n matching the pattern (1,2,3).

0, 0, 0, 1, 19, 257, 3167, 38909, 498235, 6811453, 100623211, 1612937661, 28033056683, 526501880989, 10639153638795, 230269650097469, 5315570416909995, 130370239796988957, 3385531348514480651, 92801566389186549245, 2677687663571344712043, 81124824154544921317597

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

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

Formula

a(n) = A000670(n) - A226316(n). - Andrew Howroyd, Jan 28 2024

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.

Compositions matching this pattern are counted by A335514 and ranked by A335479.

Permutations of prime indices matching this pattern are counted by A335520.

Patterns are counted by A000670 and ranked by A333217.

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.

Cf. A034691, A158005, A158009, A238279, A333218, A333379, A333942.

Sequence in context: A021154 A255722 A016313 * A017917 A016308 A014923

Adjacent sequences: A335512 A335513 A335514 * A335516 A335517 A335518

Keyword

nonn

Author

Gus Wiseman, Jun 19 2020

Extensions

a(9) onwards from Andrew Howroyd, Jan 28 2024

Status

approved