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%I #9 Jun 30 2020 09:55:26
%S 5,9,11,13,17,18,19,21,22,23,25,27,29,33,34,35,37,38,39,41,43,44,45,
%T 46,47,49,50,51,53,54,55,57,59,61,65,66,67,68,69,70,71,73,74,75,76,77,
%U 78,79,81,82,83,85,86,87,88,89,90,91,92,93,94,95,97,98,99
%N Numbers k such that the k-th composition in standard order (A066099) is not weakly increasing.
%C Also compositions matching the pattern (2,1).
%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.
%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 5: (2,1)
%e 9: (3,1)
%e 11: (2,1,1)
%e 13: (1,2,1)
%e 17: (4,1)
%e 18: (3,2)
%e 19: (3,1,1)
%e 21: (2,2,1)
%e 22: (2,1,2)
%e 23: (2,1,1,1)
%e 25: (1,3,1)
%e 27: (1,2,1,1)
%e 29: (1,1,2,1)
%e 33: (5,1)
%e 34: (4,2)
%e 35: (4,1,1)
%t stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
%t Select[Range[0,100],MatchQ[stc[#],{___,x_,___,y_,___}/;x>y]&]
%Y The complement A225620 is the avoiding version.
%Y The (1,2)-matching version is A335485.
%Y Patterns matching this pattern are counted by A002051 (by length).
%Y Permutations of prime indices matching this pattern are counted by A008480(n) - 1.
%Y These compositions are counted by A056823 (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 Cf. A034691, A056986, A108917, A114994, A238279, A334968, A335456, A335458.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jun 18 2020
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