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A374684 Sum of leaders of strictly increasing runs in the n-th composition in standard order.

%I A374684 #6 Jul 27 2024 15:57:10
%S A374684 0,1,2,2,3,3,1,3,4,4,4,4,1,2,2,4,5,5,5,5,2,5,3,5,1,2,3,3,2,3,3,5,6,6,
%T A374684 6,6,6,6,4,6,2,3,6,6,3,4,4,6,1,2,3,3,1,4,2,4,2,3,4,4,3,4,4,6,7,7,7,7,
%U A374684 7,7,5,7,3,7,7,7,4,5,5,7,2,3,4,4,4,7,5
%N A374684 Sum of leaders of strictly increasing runs in the n-th composition in standard order.
%C A374684 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.
%H A374684 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374684 The maximal strictly increasing subsequences of the 1234567th composition in standard order are ((3),(2),(1,2),(2),(1,2,5),(1),(1),(1)) with leaders (3,2,1,2,1,1,1,1), so a(1234567) = 12.
%t A374684 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A374684 Table[Total[First/@Split[stc[n],Less]],{n,0,100}]
%Y A374684 The weak version is A374630.
%Y A374684 Row-sums of A374683.
%Y A374684 The opposite version is A374758.
%Y A374684 All of the following pertain to compositions in standard order:
%Y A374684 - Length is A000120.
%Y A374684 - Sum is A029837(n+1) (or sometimes A070939).
%Y A374684 - Parts are listed by A066099.
%Y A374684 - Number of adjacent equal pairs is A124762, unequal A333382.
%Y A374684 - Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
%Y A374684 - Run-length transform is A333627.
%Y A374684 - Run-compression transform is A373948.
%Y A374684 - Ranks of contiguous compositions are A374249, counted by A274174.
%Y A374684 - Ranks of non-contiguous compositions are A374253, counted by A335548.
%Y A374684 Cf. A065120, A106356, A188920, A189076, A238343, A272919, A374634, A374685, A374698.
%Y A374684 Cf. A374251 (sums A373953), A374515 (sums A374516), A374740 (sums A374741).
%K A374684 nonn
%O A374684 0,3
%A A374684 _Gus Wiseman_, Jul 26 2024