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A335517
Number of matching pairs of patterns, the longest having length n.
4
1, 2, 9, 64, 623, 7866, 122967
OFFSET
0,2
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).
EXAMPLE
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)
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}]
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.
Sequence in context: A141209 A269770 A269649 * A128577 A052514 A274395
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jun 23 2020
STATUS
approved