%I #14 Jun 29 2020 22:15:13
%S 1,2,5,12,32,84,211,556,1446,3750,9824,25837,67681,178160,468941,
%T 1233837,3248788
%N Number of normal patterns matched by compositions of n.
%C A composition of n is a finite sequence of positive integers summing to n.
%C 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).
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>
%H Gus Wiseman, <a href="https://oeis.org/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>
%e The 8 compositions of 4 together with the a(4) = 32 patterns they match:
%e 4: 31: 13: 22: 211: 121: 112: 1111:
%e -----------------------------------------------------
%e () () () () () () () ()
%e (1) (1) (1) (1) (1) (1) (1) (1)
%e (21) (12) (11) (11) (11) (11) (11)
%e (21) (12) (12) (111)
%e (211) (21) (112) (1111)
%e (121)
%t mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
%t Table[Sum[Length[Union[mstype/@Subsets[y]]],{y,Join@@Permutations/@IntegerPartitions[n]}],{n,0,8}]
%Y References found in the link are not all included here.
%Y The version for standard compositions is A335454.
%Y The contiguous case is A335457.
%Y The version for Heinz numbers of partitions is A335549.
%Y Patterns are counted by A000670 and ranked by A333217.
%Y The n-th composition has A124771(n) distinct consecutive subsequences.
%Y Knapsack compositions are counted by A325676 and ranked by A333223.
%Y The n-th composition has A333257(n) distinct subsequence-sums.
%Y The n-th composition has A334299(n) distinct subsequences.
%Y Minimal patterns avoided by a standard composition are counted by A335465.
%Y Cf. A034691, A056986, A106356, A108917, A124770, A269134, A329744, A333224, A335458.
%K nonn,more
%O 0,2
%A _Gus Wiseman_, Jun 16 2020
%E a(14)-a(16) from _Jinyuan Wang_, Jun 26 2020