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%I #9 Jun 30 2020 09:56:07
%S 7,15,23,27,29,30,31,39,42,47,51,55,57,59,60,61,62,63,71,79,85,86,87,
%T 90,91,93,94,95,99,103,106,107,109,110,111,113,115,117,118,119,120,
%U 121,122,123,124,125,126,127,135,143,151,155,157,158,159,167,170,171
%N Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,1,1).
%C These are compositions with some part appearing more than twice.
%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 The sequence of terms together with the corresponding compositions begins:
%e 7: (1,1,1)
%e 15: (1,1,1,1)
%e 23: (2,1,1,1)
%e 27: (1,2,1,1)
%e 29: (1,1,2,1)
%e 30: (1,1,1,2)
%e 31: (1,1,1,1,1)
%e 39: (3,1,1,1)
%e 42: (2,2,2)
%e 47: (2,1,1,1,1)
%e 51: (1,3,1,1)
%e 55: (1,2,1,1,1)
%e 57: (1,1,3,1)
%e 59: (1,1,2,1,1)
%e 60: (1,1,1,3)
%t stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
%t Select[Range[0,100],MatchQ[stc[#],{___,x_,___,x_,___,x_,___}]&]
%Y The complement A335513 is the avoiding version.
%Y Patterns matching this pattern are counted by A335508 (by length).
%Y Permutations of prime indices matching this pattern are counted by A335510.
%Y These compositions are counted by A335455 (by sum).
%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.
%Y Patterns matched by standard compositions are counted by A335454.
%Y Minimal patterns avoided by a standard composition are counted by A335465.
%Y The (1,1)-matching version is A335488.
%Y Cf. A034691, A056986, A114994, A238279, A334968, A335456, A335458.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jun 18 2020
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