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A335483 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (3,1,2).

%I A335483 #9 Jun 29 2020 22:13:26
%S A335483 38,70,77,78,102,134,140,141,142,150,154,155,157,158,166,198,205,206,
%T A335483 230,262,268,269,270,276,278,281,282,283,284,285,286,294,301,302,306,
%U A335483 308,309,310,311,314,315,317,318,326,333,334,358,390,396,397,398,406,410
%N A335483 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (3,1,2).
%C A335483 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 A335483 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 A335483 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>
%H A335483 Gus Wiseman, <a href="https://oeis.org/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>
%H A335483 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 A335483 The sequence of terms together with the corresponding compositions begins:
%e A335483    38: (3,1,2)
%e A335483    70: (4,1,2)
%e A335483    77: (3,1,2,1)
%e A335483    78: (3,1,1,2)
%e A335483   102: (1,3,1,2)
%e A335483   134: (5,1,2)
%e A335483   140: (4,1,3)
%e A335483   141: (4,1,2,1)
%e A335483   142: (4,1,1,2)
%e A335483   150: (3,2,1,2)
%e A335483   154: (3,1,2,2)
%e A335483   155: (3,1,2,1,1)
%e A335483   157: (3,1,1,2,1)
%e A335483   158: (3,1,1,1,2)
%e A335483   166: (2,3,1,2)
%t A335483 stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
%t A335483 Select[Range[0,100],MatchQ[stc[#],{___,x_,___,y_,___,z_,___}/;y<z<x]&]
%Y A335483 The version counting permutations is A056986.
%Y A335483 Patterns matching this pattern are counted by A335515 (by length).
%Y A335483 Permutations of prime indices matching this pattern are counted by A335520.
%Y A335483 These compositions are counted by A335514 (by sum).
%Y A335483 Constant patterns are counted by A000005 and ranked by A272919.
%Y A335483 Permutations are counted by A000142 and ranked by A333218.
%Y A335483 Patterns are counted by A000670 and ranked by A333217.
%Y A335483 Non-unimodal compositions are counted by A115981 and ranked by A335373.
%Y A335483 Permutations matching (1,3,2,4) are counted by A158009.
%Y A335483 Combinatory separations are counted by A269134.
%Y A335483 Patterns matched by standard compositions are counted by A335454.
%Y A335483 Minimal patterns avoided by a standard composition are counted by A335465.
%Y A335483 Other permutations:
%Y A335483 - A335479 (1,2,3)
%Y A335483 - A335480 (1,3,2)
%Y A335483 - A335481 (2,1,3)
%Y A335483 - A335482 (2,3,1)
%Y A335483 - A335483 (3,1,2)
%Y A335483 - A335484 (3,2,1)
%Y A335483 Cf. A034691, A056986, A108917, A114994, A158005, A238279, A333224, A333257, A334968, A335456, A335458.
%K A335483 nonn
%O A335483 1,1
%A A335483 _Gus Wiseman_, Jun 18 2020