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A374685 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are identical.

%I A374685 #5 Jul 27 2024 15:57:05
%S A374685 0,1,2,3,4,6,7,8,10,12,13,14,15,16,20,24,25,27,28,29,30,31,32,36,40,
%T A374685 42,48,49,51,52,54,55,56,57,59,60,61,62,63,64,72,80,82,84,96,97,99,
%U A374685 102,103,104,105,108,109,110,111,112,113,115,116,118,119,120,121
%N A374685 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are identical.
%C A374685 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 A374685 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 A374685 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374685 The maximal strictly increasing subsequences of the 6560th composition in standard order are ((1,3),(1,2,6)), with leaders (1,1), so 6560 is in the sequence.
%e A374685 The terms together with corresponding compositions begin:
%e A374685    0: ()
%e A374685    1: (1)
%e A374685    2: (2)
%e A374685    3: (1,1)
%e A374685    4: (3)
%e A374685    6: (1,2)
%e A374685    7: (1,1,1)
%e A374685    8: (4)
%e A374685   10: (2,2)
%e A374685   12: (1,3)
%e A374685   13: (1,2,1)
%e A374685   14: (1,1,2)
%e A374685   15: (1,1,1,1)
%e A374685   16: (5)
%e A374685   20: (2,3)
%e A374685   24: (1,4)
%e A374685   25: (1,3,1)
%e A374685   27: (1,2,1,1)
%e A374685   28: (1,1,3)
%e A374685   29: (1,1,2,1)
%e A374685   30: (1,1,1,2)
%e A374685   31: (1,1,1,1,1)
%t A374685 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A374685 Select[Range[0,100],SameQ@@First/@Split[stc[#],Less]&]
%Y A374685 The weak version is A374633, counted by A374631.
%Y A374685 Positions of constant rows in A374683.
%Y A374685 Compositions of this type are counted by A374686.
%Y A374685 For distinct leaders we have A374698, counted by A374687.
%Y A374685 The opposite version is A374759, counted by A374760.
%Y A374685 Other types of runs: A272919 (counts A000005), A374519 (counts A374517), A374744 (counts A374742).
%Y A374685 A011782 counts compositions.
%Y A374685 A238130, A238279, A333755 count compositions by number of runs.
%Y A374685 A374748 counts compositions by sum of leaders of weakly decreasing runs.
%Y A374685 All of the following pertain to compositions in standard order:
%Y A374685 - Length is A000120.
%Y A374685 - Sum is A029837(n+1) (or sometimes A070939).
%Y A374685 - Parts are listed by A066099.
%Y A374685 - Adjacent equal pairs are counted by A124762, unequal A333382.
%Y A374685 - Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
%Y A374685 - Ranks of anti-run compositions are A333489, counted by A003242.
%Y A374685 - Run-length transform is A333627.
%Y A374685 - Run-compression transform is A373948, sum A373953, excess A373954.
%Y A374685 - Ranks of contiguous compositions are A374249, counted by A274174.
%Y A374685 Cf. A065120, A106356, A238343, A333213, A373949, A374520, A374629, A374630, A374701, A374740, A374768.
%K A374685 nonn
%O A374685 1,3
%A A374685 _Gus Wiseman_, Jul 27 2024