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A335480 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,3,2).

%I A335480 #9 Jun 29 2020 22:11:51
%S A335480 50,98,101,102,114,178,194,196,197,198,202,203,205,206,210,226,229,
%T A335480 230,242,306,324,354,357,358,370,386,388,389,390,393,394,395,396,397,
%U A335480 398,402,404,405,406,407,410,411,413,414,418,421,422,434,450,452,453,454
%N A335480 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,3,2).
%C A335480 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 A335480 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 A335480 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>
%H A335480 Gus Wiseman, <a href="https://oeis.org/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>
%H A335480 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 A335480 The sequence of terms together with the corresponding compositions begins:
%e A335480    50: (1,3,2)
%e A335480    98: (1,4,2)
%e A335480   101: (1,3,2,1)
%e A335480   102: (1,3,1,2)
%e A335480   114: (1,1,3,2)
%e A335480   178: (2,1,3,2)
%e A335480   194: (1,5,2)
%e A335480   196: (1,4,3)
%e A335480   197: (1,4,2,1)
%e A335480   198: (1,4,1,2)
%e A335480   202: (1,3,2,2)
%e A335480   203: (1,3,2,1,1)
%e A335480   205: (1,3,1,2,1)
%e A335480   206: (1,3,1,1,2)
%e A335480   210: (1,2,3,2)
%t A335480 stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
%t A335480 Select[Range[0,100],MatchQ[stc[#],{___,x_,___,y_,___,z_,___}/;x<z<y]&]
%Y A335480 The version counting permutations is A056986.
%Y A335480 Patterns matching this pattern are counted by A335515 (by length).
%Y A335480 Permutations of prime indices matching this pattern are counted by A335520.
%Y A335480 These compositions are counted by A335514 (by sum).
%Y A335480 Constant patterns are counted by A000005 and ranked by A272919.
%Y A335480 Permutations are counted by A000142 and ranked by A333218.
%Y A335480 Patterns are counted by A000670 and ranked by A333217.
%Y A335480 Non-unimodal compositions are counted by A115981 and ranked by A335373.
%Y A335480 Permutations matching (1,3,2,4) are counted by A158009.
%Y A335480 Combinatory separations are counted by A269134.
%Y A335480 Patterns matched by standard compositions are counted by A335454.
%Y A335480 Minimal patterns avoided by a standard composition are counted by A335465.
%Y A335480 Other permutations:
%Y A335480 - A335479 (1,2,3)
%Y A335480 - A335480 (1,3,2)
%Y A335480 - A335481 (2,1,3)
%Y A335480 - A335482 (2,3,1)
%Y A335480 - A335483 (3,1,2)
%Y A335480 - A335484 (3,2,1)
%Y A335480 Cf. A034691, A056986, A108917, A114994, A158005, A238279, A333224, A333257, A334968, A335456, A335458.
%K A335480 nonn
%O A335480 1,1
%A A335480 _Gus Wiseman_, Jun 18 2020