A374684 - OEIS

%I #6 Jul 27 2024 15:57:10

%S 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 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 7,7,5,7,3,7,7,7,4,5,5,7,2,3,4,4,4,7,5

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

%C 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 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.

%e 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 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t Table[Total[First/@Split[stc[n],Less]],{n,0,100}]

%Y The weak version is A374630.

%Y Row-sums of A374683.

%Y The opposite version is A374758.

%Y All of the following pertain to compositions in standard order:

%Y - Length is A000120.

%Y - Sum is A029837(n+1) (or sometimes A070939).

%Y - Parts are listed by A066099.

%Y - Number of adjacent equal pairs is A124762, unequal A333382.

%Y - Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.

%Y - Run-length transform is A333627.

%Y - Run-compression transform is A373948.

%Y - Ranks of contiguous compositions are A374249, counted by A274174.

%Y - Ranks of non-contiguous compositions are A374253, counted by A335548.

%Y Cf. A065120, A106356, A188920, A189076, A238343, A272919, A374634, A374685, A374698.

%Y Cf. A374251 (sums A373953), A374515 (sums A374516), A374740 (sums A374741).

%K nonn

%O 0,3

%A _Gus Wiseman_, Jul 26 2024