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Number of normal patterns matched by the n-th composition in standard order (A066099).
51

%I #14 Jun 29 2020 22:15:28

%S 1,2,2,3,2,3,3,4,2,3,3,5,3,6,5,5,2,3,3,5,3,5,6,7,3,6,5,9,5,9,7,6,2,3,

%T 3,5,3,4,5,7,3,5,4,7,5,10,9,9,3,6,5,9,4,9,10,12,5,9,7,13,7,12,9,7,2,3,

%U 3,5,3,4,5,7,3,5,5,7,6,10,9,9,3,5,6,8,5

%N Number of normal patterns matched by the n-th composition in standard order (A066099).

%C We define a (normal) pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670. 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).

%C The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>

%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</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 a(n) patterns for n = 0, 1, 3, 7, 11, 13, 23, 83, 27, 45:

%e 0: 1: 11: 111: 211: 121: 2111: 2311: 1211: 2121:

%e ---------------------------------------------------------------------

%e () () () () () () () () () ()

%e (1) (1) (1) (1) (1) (1) (1) (1) (1)

%e (11) (11) (11) (11) (11) (11) (11) (11)

%e (111) (21) (12) (21) (12) (12) (12)

%e (211) (21) (111) (21) (21) (21)

%e (121) (211) (211) (111) (121)

%e (2111) (231) (121) (211)

%e (2311) (211) (212)

%e (1211) (221)

%e (2121)

%t stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];

%t mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];

%t Table[Length[Union[mstype/@Subsets[stc[n]]]],{n,0,30}]

%Y References found in the links are not all included here.

%Y Summing over indices with binary length n gives A335456(n).

%Y The contiguous case is A335458.

%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 avoided patterns are counted by A335465.

%Y Cf. A034691, A056986, A108917, A124767, A124770, A158005, A269134, A333218, A333222, A333224, A334030.

%K nonn

%O 0,2

%A _Gus Wiseman_, Jun 14 2020