A335508 Number of patterns of length n matching the pattern (1,1,1).

0, 0, 0, 1, 9, 91, 993, 12013, 160275, 2347141, 37496163, 649660573, 12142311195, 243626199181, 5224710549243, 119294328993853, 2889836999693355, 74037381200415901, 2000383612949821323, 56850708386783835133, 1695491518035158123115, 52949018580275965241821

Offset: 0

Author

Gus Wiseman, Jun 18 2020

Keywords

• nonn

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).

Examples

			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)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..424
• Wikipedia, Permutation pattern
• Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Crossrefs

The complement A080599 is the avoiding version.

Permutations of prime indices matching this pattern are counted by A335510.

Compositions matching this pattern are counted by A335455 and ranked by A335512.

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, A232464, A238279, A292884, A333175, A333755, A335451, A335456, A335457, A335458.

Cf. A276922.

Programs

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}]

Formula

a(n) = Sum_{k=3..n} A276922(n,k). - Alois P. Heinz, Jan 28 2024

a(n) = A000670(n) - A080599(n). - Andrew Howroyd, Jan 28 2024

Extensions

a(9)-a(21) from Alois P. Heinz, Jan 28 2024