A335518 -id:A335518 - cp's OEIS frontend

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

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

Offset: 0

Gus Wiseman, Jun 23 2020

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).

			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)

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

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.

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}]

cp's OEIS Frontend

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

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