cp's OEIS Frontend

A335478 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (2,1,1).

%I A335478 #9 Jun 30 2020 09:55:20
%S A335478 11,19,23,27,35,39,43,45,46,47,51,55,59,67,71,74,75,77,78,79,83,87,89,
%T A335478 91,92,93,94,95,99,103,107,109,110,111,115,119,123,131,135,138,139,
%U A335478 141,142,143,147,149,150,151,153,154,155,156,157,158,159,163,167,171
%N A335478 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (2,1,1).
%C A335478 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 A335478 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 A335478 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>
%H A335478 Gus Wiseman, <a href="https://oeis.org/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>
%H A335478 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 A335478 The sequence of terms together with the corresponding compositions begins:
%e A335478   11: (2,1,1)
%e A335478   19: (3,1,1)
%e A335478   23: (2,1,1,1)
%e A335478   27: (1,2,1,1)
%e A335478   35: (4,1,1)
%e A335478   39: (3,1,1,1)
%e A335478   43: (2,2,1,1)
%e A335478   45: (2,1,2,1)
%e A335478   46: (2,1,1,2)
%e A335478   47: (2,1,1,1,1)
%e A335478   51: (1,3,1,1)
%e A335478   55: (1,2,1,1,1)
%e A335478   59: (1,1,2,1,1)
%e A335478   67: (5,1,1)
%e A335478   71: (4,1,1,1)
%t A335478 stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
%t A335478 Select[Range[0,100],MatchQ[stc[#],{___,x_,___,y_,___,y_,___}/;x>y]&]
%Y A335478 The complement A335523 is the avoiding version.
%Y A335478 The (1,1,2)-matching version is A335476.
%Y A335478 Patterns matching this pattern are counted by A335509 (by length).
%Y A335478 Permutations of prime indices matching this pattern are counted by A335516.
%Y A335478 These compositions are counted by A335470 (by sum).
%Y A335478 Constant patterns are counted by A000005 and ranked by A272919.
%Y A335478 Permutations are counted by A000142 and ranked by A333218.
%Y A335478 Patterns are counted by A000670 and ranked by A333217.
%Y A335478 Non-unimodal compositions are counted by A115981 and ranked by A335373.
%Y A335478 Combinatory separations are counted by A269134.
%Y A335478 Patterns matched by standard compositions are counted by A335454.
%Y A335478 Minimal patterns avoided by a standard composition are counted by A335465.
%Y A335478 Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335446, A335456, A335458, A335475.
%K A335478 nonn
%O A335478 1,1
%A A335478 _Gus Wiseman_, Jun 18 2020