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A374633 Numbers k such that the leaders of weakly increasing runs in the k-th composition in standard order (A066099) are identical.

%I A374633 #9 Jul 22 2024 21:11:16
%S A374633 0,1,2,3,4,6,7,8,10,12,13,14,15,16,20,24,25,26,27,28,29,30,31,32,36,
%T A374633 40,42,48,49,51,52,53,54,55,56,57,58,59,60,61,62,63,64,72,80,82,84,96,
%U A374633 97,99,100,102,103,104,105,106,107,108,109,110,111,112,113,115
%N A374633 Numbers k such that the leaders of weakly increasing runs in the k-th composition in standard order (A066099) are identical.
%C A374633 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 A374633 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 A374633 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374633 The maximal weakly increasing subsequences of the 26165th composition in standard order are ((1,3),(1,4),(1,2,2),(1)), with leaders (1,1,1,1), so 26165 is in the sequence.
%e A374633 The sequence together with the corresponding compositions begins:
%e A374633    0: ()
%e A374633    1: (1)
%e A374633    2: (2)
%e A374633    3: (1,1)
%e A374633    4: (3)
%e A374633    6: (1,2)
%e A374633    7: (1,1,1)
%e A374633    8: (4)
%e A374633   10: (2,2)
%e A374633   12: (1,3)
%e A374633   13: (1,2,1)
%e A374633   14: (1,1,2)
%e A374633   15: (1,1,1,1)
%e A374633   16: (5)
%e A374633   20: (2,3)
%e A374633   24: (1,4)
%e A374633   25: (1,3,1)
%e A374633   26: (1,2,2)
%e A374633   27: (1,2,1,1)
%t A374633 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A374633 Select[Range[0,100],SameQ@@First/@Split[stc[#],LessEqual]&]
%Y A374633 For strictly decreasing leaders we appear to have A188920.
%Y A374633 For weakly decreasing leaders we appear to have A189076.
%Y A374633 Other types of runs: A272919 (counted by A000005), A374519 (counted by A374517), A374685 (counted by A374686), A374744 (counted by A374742), A374759 (counted by A374760).
%Y A374633 Positions of constant rows in A374629 (which has sums A374630).
%Y A374633 Compositions of this type are counted by A374631.
%Y A374633 For strictly increasing leaders see A374634.
%Y A374633 For all different leaders we have A374768, counted by A374632.
%Y A374633 A011782 counts compositions.
%Y A374633 A238130, A238279, A333755 count compositions by number of runs.
%Y A374633 A374637 counts compositions by sum of leaders of weakly increasing runs.
%Y A374633 All of the following pertain to compositions in standard order:
%Y A374633 - Ones are counted by A000120.
%Y A374633 - Sum is A029837 (or sometimes A070939).
%Y A374633 - Parts are listed by A066099.
%Y A374633 - Length is A070939.
%Y A374633 - Adjacent equal pairs are counted by A124762, unequal A333382.
%Y A374633 - Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
%Y A374633 - Ranks of anti-run compositions are A333489, counted by A003242.
%Y A374633 - Run-length transform is A333627.
%Y A374633 - Run-compression transform is A373948, sum A373953, excess A373954.
%Y A374633 - Ranks of contiguous compositions are A374249, counted by A274174.
%Y A374633 Cf. A065120, A106356, A238343, A333175, A333213, A373949, A374635.
%K A374633 nonn
%O A374633 1,3
%A A374633 _Gus Wiseman_, Jul 21 2024