A335517 Number of matching pairs of patterns, the longest having length n.

1, 2, 9, 64, 623, 7866, 122967

Offset: 0

Author

Gus Wiseman, Jun 23 2020

Keywords

• nonn
• more

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(0) = 1 through a(2) = 9 pairs of patterns:
  ()<=()    ()<=(1)      ()<=(1,1)
           (1)<=(1)      ()<=(1,2)
                         ()<=(2,1)
                        (1)<=(1,1)
                        (1)<=(1,2)
                        (1)<=(2,1)
                      (1,1)<=(1,1)
                      (1,2)<=(1,2)
                      (2,1)<=(2,1)
		

Links

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

Crossrefs

Row sums of A335518.

Patterns are counted by A000670 and ranked by A333217.

Patterns matched by a standard composition are counted by A335454.

Patterns contiguously matched by compositions are counted by A335457.

Minimal patterns avoided by a standard composition are counted by A335465.

Patterns matched by prime indices are counted by A335549.

Cf. A011782, A034691, A056986, A124771, A269134, A329744, A333257, A334299.

Programs

Mathematica

mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Sum[Length[Union[mstype/@Subsets[y]]],{y,Join@@Permutations/@allnorm[n]}],{n,0,5}]