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A374698 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are distinct.

%I A374698 #5 Jul 27 2024 15:57:00
%S A374698 0,1,2,4,5,6,8,9,12,16,17,18,20,22,24,26,32,33,34,37,38,40,41,44,48,
%T A374698 50,52,64,65,66,68,69,70,72,76,80,81,88,96,98,100,104,128,129,130,132,
%U A374698 133,134,137,140,144,145,148,150,152,154,160,161,164,166,176,180
%N A374698 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are distinct.
%C A374698 The leaders of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.
%C A374698 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 A374698 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374698 The maximal strictly increasing subsequences of the 212th composition in standard order are ((1,2),(2,3)), with leaders (1,2), so 212 is in the sequence.
%e A374698 The terms together with corresponding compositions begin:
%e A374698    0: ()
%e A374698    1: (1)
%e A374698    2: (2)
%e A374698    4: (3)
%e A374698    5: (2,1)
%e A374698    6: (1,2)
%e A374698    8: (4)
%e A374698    9: (3,1)
%e A374698   12: (1,3)
%e A374698   16: (5)
%e A374698   17: (4,1)
%e A374698   18: (3,2)
%e A374698   20: (2,3)
%e A374698   22: (2,1,2)
%e A374698   24: (1,4)
%e A374698   26: (1,2,2)
%t A374698 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A374698 Select[Range[0,100],UnsameQ@@First/@Split[stc[#],Less]&]
%Y A374698 Positions of distinct (strict) rows in A374683.
%Y A374698 For identical leaders we have A374685, counted by A374761.
%Y A374698 Compositions of this type are counted by A374687.
%Y A374698 The opposite version is A374767, counted by A374760.
%Y A374698 The weak version is A374768, counted by A374632.
%Y A374698 Other types of runs: A374249 (counts A274174), A374638 (counts A374518), A374701 (counts A374743).
%Y A374698 A011782 counts compositions.
%Y A374698 A238130, A238279, A333755 count compositions by number of runs.
%Y A374698 All of the following pertain to compositions in standard order:
%Y A374698 - Length is A000120.
%Y A374698 - Sum is A029837(n+1) (or sometimes A070939).
%Y A374698 - Parts are listed by A066099.
%Y A374698 - Adjacent equal pairs are counted by A124762, unequal A333382.
%Y A374698 - Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
%Y A374698 - Ranks of anti-run compositions are A333489, counted by A003242.
%Y A374698 - Run-length transform is A333627.
%Y A374698 - Run-compression transform is A373948, sum A373953, excess A373954.
%Y A374698 Cf. A065120, A106356, A238343, A333213, A373949, A374520, A374629, A374630, A374635, A374740.
%K A374698 nonn
%O A374698 1,3
%A A374698 _Gus Wiseman_, Jul 27 2024