A335469 - OEIS

%I #11 Sep 19 2024 08:27:56

%S 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,23,24,25,26,

%T 27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,47,48,49,50,51,

%U 52,53,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71

%N Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (2,1,2).

%C First differs from A374701 in having 150, corresponding to (3,2,1,2). - _Gus Wiseman_, Sep 18 2024

%C A composition of n is a finite sequence of positive integers summing to n. 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 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>

%e See A335468 for an example of the complement.

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

%t Select[Range[0,100],!MatchQ[stc[#],{___,x_,___,y_,___,x_,___}/;x>y]&]

%Y The complement A335468 is the matching version.

%Y The (1,2,1)-avoiding version is A335467.

%Y These compositions are counted by A335473.

%Y Constant patterns are counted by A000005 and ranked by A272919.

%Y Permutations are counted by A000142 and ranked by A333218.

%Y Patterns are counted by A000670 and ranked by A333217.

%Y Non-unimodal compositions are counted by A115981 and ranked by A335373.

%Y Combinatory separations are counted by A269134 and ranked by A334030.

%Y Patterns matched by standard compositions are counted by A335454.

%Y Minimal patterns avoided by a standard composition are counted by A335465.

%Y Cf. A001710, A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335450, A335456, A335458.

%K nonn

%O 1,3

%A _Gus Wiseman_, Jun 16 2020