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Number of normal patterns matched by compositions of n.
88

%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