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A374767 Numbers k such that the leaders of strictly decreasing runs in the k-th composition in standard order are distinct.

%I A374767 #6 Jul 31 2024 09:08:56
%S A374767 0,1,2,4,5,6,8,9,11,12,13,16,17,18,19,20,24,25,32,33,34,35,37,38,40,
%T A374767 41,44,48,49,50,52,64,65,66,67,68,69,70,72,74,75,77,78,80,81,82,83,88,
%U A374767 89,92,96,97,98,101,102,104,105,108,128,129,130,131,132,133
%N A374767 Numbers k such that the leaders of strictly decreasing runs in the k-th composition in standard order are distinct.
%C A374767 The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.
%C A374767 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 A374767 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374767 The 10000000th composition in standard order is (3,1,4,3,2,1,2,8), with strictly decreasing runs ((3,1),(4,3,2,1),(2),(8)), with leaders (3,4,2,1) so 10000000 is in the sequence.
%e A374767 The terms together with the corresponding compositions begin:
%e A374767    0: ()
%e A374767    1: (1)
%e A374767    2: (2)
%e A374767    4: (3)
%e A374767    5: (2,1)
%e A374767    6: (1,2)
%e A374767    8: (4)
%e A374767    9: (3,1)
%e A374767   11: (2,1,1)
%e A374767   12: (1,3)
%e A374767   13: (1,2,1)
%e A374767   16: (5)
%e A374767   17: (4,1)
%e A374767   18: (3,2)
%e A374767   19: (3,1,1)
%e A374767   20: (2,3)
%e A374767   24: (1,4)
%e A374767   25: (1,3,1)
%t A374767 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A374767 Select[Range[0,100],UnsameQ@@First/@Split[stc[#],Greater]&]
%Y A374767 The opposite version is A374698, counted by A374687.
%Y A374767 The weak version is A374701, counted by A374743.
%Y A374767 For identical instead of distinct runs we have A374759, counted by A374760.
%Y A374767 Compositions of this type are counted by A374761.
%Y A374767 All of the following pertain to compositions in standard order:
%Y A374767 - Length is A000120.
%Y A374767 - Sum is A029837(n+1).
%Y A374767 - Parts are listed by A066099.
%Y A374767 - Number of adjacent equal pairs is A124762, unequal A333382.
%Y A374767 - Run-length transform is A333627, sum A070939.
%Y A374767 - Run-compression transform is A373948, sum A373953, excess A373954.
%Y A374767 - Ranks of contiguous compositions are A374249, counted by A274174.
%Y A374767 Six types of runs:
%Y A374767 - Count: A124766, A124765, A124768, A124769, A333381, A124767.
%Y A374767 - Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
%Y A374767 - Rank: A375123, A375124, A375125, A375126, A375127, A373948.
%Y A374767 Cf. A065120, A106356, A188920, A233564, A238343, A272919, A333213, A373949, A374633, A374685, A374744, A374758, A375128.
%K A374767 nonn
%O A374767 1,3
%A A374767 _Gus Wiseman_, Jul 29 2024