A335458 - OEIS

%I #11 Jun 29 2020 22:15:40

%S 1,2,2,3,2,3,3,4,2,3,3,5,3,5,5,5,2,3,3,5,3,5,5,7,3,5,5,8,5,8,7,6,2,3,

%T 3,5,3,4,5,7,3,5,4,7,5,7,8,9,3,5,5,8,4,8,7,11,5,8,7,11,7,11,9,7,2,3,3,

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

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

%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.

%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 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>

%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>

%F a(n) = A335474(n) + 1.

%e The a(180) = 7 patterns are: (), (1), (1,2), (2,1), (1,2,3), (2,1,2), (2,1,2,3).

%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/@ReplaceList[stc[n],{___,s___,___}:>{s}]]],{n,0,30}]

%Y The non-contiguous version is A335454.

%Y Summing over indices with binary length n gives A335457(n).

%Y The nonempty version is A335474.

%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. A108917, A124767, A124770, A181796, A269134, A333218, A333222, A333628, A334030, A335456.

%K nonn

%O 0,2

%A _Gus Wiseman_, Jun 21 2020