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%I #10 Jun 27 2020 22:55:57
%S 1,2,5,9,18,31,54,89,145,225,349,524,778,1137,1645,2330,3293,4586,
%T 6341,8676,11794,15880,21292,28298,37419,49163,64301,83576,108191,
%U 139326,178699,228183,290286,367760,464374,584146,732481,915468,1140773,1417115,1755578
%N Number of normal patterns contiguously matched by integer partitions of n.
%C We define a (normal) 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 contiguously match a pattern P if there is a contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) contiguously matches (1,1,2) and (2,1,1) but not (2,1,2), (1,2,1), (1,2,2), or (2,2,1).
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>
%H Gus Wiseman, <a href="/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>
%e The patterns contiguously matched by (3,2,2,1) are: (), (1), (1,1), (2,1), (2,1,1), (2,2,1), (3,2,2,1). Note that (3,2,1) is not contiguously matched. See A335837 for a larger example.
%t mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
%t Table[Sum[Length[Union[mstype/@ReplaceList[y,{___,s___,___}:>{s}]]],{y,IntegerPartitions[n]}],{n,0,8}]
%Y The version for compositions in standard order is A335474.
%Y The version for compositions is A335457.
%Y The not necessarily contiguous version is A335837.
%Y Patterns are counted by A000670 and ranked by A333217.
%Y Patterns contiguously matched by prime indices are counted by A335516.
%Y Contiguous divisors are counted by A335519.
%Y Minimal patterns avoided by prime indices are counted by A335550.
%Y Cf. A056986, A108917, A333257, A335454, A335456, A335458, A335549.
%K nonn
%O 0,2
%A _Gus Wiseman_, Jun 27 2020
%E More terms from _Jinyuan Wang_, Jun 27 2020
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