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%I #14 Nov 30 2022 11:57:31
%S 6,12,13,14,20,22,24,25,26,27,28,29,30,38,40,41,44,45,46,48,49,50,51,
%T 52,53,54,55,56,57,58,59,60,61,62,70,72,76,77,78,80,81,82,83,84,86,88,
%U 89,90,91,92,93,94,96,97,98,99,100,101,102,103,104,105,106
%N Numbers k such that the k-th composition in standard order (A066099) is not weakly decreasing.
%C Also compositions matching the pattern (1,2).
%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 Keiichi Shigechi, <a href="https://arxiv.org/abs/2211.10958">Noncommutative crossing partitions</a>, arXiv:2211.10958 [math.CO], 2022.
%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 6: (1,2)
%e 12: (1,3)
%e 13: (1,2,1)
%e 14: (1,1,2)
%e 20: (2,3)
%e 22: (2,1,2)
%e 24: (1,4)
%e 25: (1,3,1)
%e 26: (1,2,2)
%e 27: (1,2,1,1)
%e 28: (1,1,3)
%e 29: (1,1,2,1)
%e 30: (1,1,1,2)
%e 38: (3,1,2)
%e 40: (2,4)
%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 A114994 is the avoiding version.
%Y The (2,1)-matching version is A335486.
%Y Patterns matching this pattern are counted by A002051 (by length).
%Y Permutations of prime indices matching this pattern are counted by A335447.
%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, A333224, A333257, A335456, A335458.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jun 18 2020
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