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A374768 Numbers k such that the leaders of weakly increasing runs in the k-th composition in standard order (A066099) are distinct.

%I A374768 #13 Jul 22 2024 21:11:11
%S A374768 0,1,2,3,4,5,6,7,8,9,10,11,12,14,15,16,17,18,19,20,21,22,23,24,26,28,
%T A374768 30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,46,47,48,50,52,56,58,60,
%U A374768 62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,78,79,80,81
%N A374768 Numbers k such that the leaders of weakly increasing runs in the k-th composition in standard order (A066099) are distinct.
%C A374768 First differs from A335467 in having 166, corresponding to the composition (2,3,1,2).
%C A374768 The leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.
%C A374768 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 A374768 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374768 The 4444th composition in standard order is (4,2,2,1,1,3), with weakly increasing runs ((4),(2,2),(1,1,3)), with leaders (4,2,1), so 4444 is in the sequence.
%t A374768 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A374768 Select[Range[0,166],UnsameQ@@First/@Split[stc[#],LessEqual]&]
%Y A374768 These are the positions of strict rows in A374629 (which has sums A374630).
%Y A374768 Compositions of this type are counted by A374632, increasing A374634.
%Y A374768 Identical instead of distinct leaders are A374633, counted by A374631.
%Y A374768 For leaders of anti-runs we have A374638, counted by A374518.
%Y A374768 For leaders of strictly increasing runs we have A374698, counted by A374687.
%Y A374768 For leaders of weakly decreasing runs we have A374701, counted by A374743.
%Y A374768 For leaders of strictly decreasing runs we have A374767, counted by A374761.
%Y A374768 A011782 counts compositions.
%Y A374768 A238130, A238279, A333755 count compositions by number of runs.
%Y A374768 All of the following pertain to compositions in standard order:
%Y A374768 - Ones are counted by A000120.
%Y A374768 - Sum is A029837 (or sometimes A070939).
%Y A374768 - Parts are listed by A066099.
%Y A374768 - Length is A070939.
%Y A374768 - Adjacent equal pairs are counted by A124762, unequal A333382.
%Y A374768 - Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
%Y A374768 - Ranks of strict compositions are A233564.
%Y A374768 - Ranks of constant compositions are A272919.
%Y A374768 - Ranks of anti-run compositions are A333489, counted by A003242.
%Y A374768 - Run-length transform is A333627.
%Y A374768 - Run-compression transform is A373948, sum A373953, excess A373954.
%Y A374768 Cf. A065120, A188920, A189076, A238343, A333213, A335449, A373949, A274174, A374249, A374635, A374637.
%K A374768 nonn
%O A374768 1,3
%A A374768 _Gus Wiseman_, Jul 19 2024