cp's OEIS Frontend

A335457 Number of normal patterns contiguously matched by compositions of n.

%I A335457 #14 Jul 08 2020 10:22:41
%S A335457 1,2,5,12,31,80,196,486,1171,2787,6564,15323,35403,81251,185087,
%T A335457 418918,942525,2109143,4695648,10405694,22959156
%N A335457 Number of normal patterns contiguously matched by compositions of n.
%C A335457 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 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).
%H A335457 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>
%H A335457 Gus Wiseman, <a href="/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>
%e A335457 The a(0) = 1 through a(3) = 12 pairs of a composition with a contiguously matched pattern:
%e A335457   ()()  (1)()   (2)()     (3)()
%e A335457         (1)(1)  (11)()    (12)()
%e A335457                 (2)(1)    (21)()
%e A335457                 (11)(1)   (3)(1)
%e A335457                 (11)(11)  (111)()
%e A335457                           (12)(1)
%e A335457                           (21)(1)
%e A335457                           (111)(1)
%e A335457                           (12)(12)
%e A335457                           (21)(21)
%e A335457                           (111)(11)
%e A335457                           (111)(111)
%t A335457 mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
%t A335457 Table[Sum[Length[Union[mstype/@ReplaceList[cmp,{___,s___,___}:>{s}]]],{cmp,Join@@Permutations/@IntegerPartitions[n]}],{n,0,10}]
%Y A335457 The version for standard compositions is A335458.
%Y A335457 The non-contiguous version is A335456.
%Y A335457 Patterns are counted by A000670 and ranked by A333217.
%Y A335457 The n-th standard composition has A124771(n) contiguous subsequences.
%Y A335457 Patterns contiguously matched by prime indices are A335549.
%Y A335457 Minimal avoided patterns of prime indices are counted by A335550.
%Y A335457 Cf. A000005, A056986, A108917, A124767, A181796, A269134, A333224, A334299.
%K A335457 nonn,more
%O A335457 0,2
%A A335457 _Gus Wiseman_, Jun 23 2020
%E A335457 a(16)-a(20) from _Jinyuan Wang_, Jul 08 2020