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Sum of leaders of strictly decreasing runs in the n-th composition in standard order.
15

%I #8 Jul 31 2024 09:09:24

%S 0,1,2,2,3,2,3,3,4,3,4,3,4,3,4,4,5,4,3,4,5,4,4,4,5,4,5,4,5,4,5,5,6,5,

%T 4,5,6,3,5,5,6,5,6,5,5,4,5,5,6,5,4,5,6,5,5,5,6,5,6,5,6,5,6,6,7,6,5,6,

%U 4,4,6,6,7,6,5,4,6,5,6,6,7,6,5,6,7,6,6

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

%C 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.

%H Gus Wiseman, <a href="/A373403/a373403.txt">Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers</a>.

%e The maximal strictly decreasing subsequences of the 1234567th composition in standard order are ((3,2,1),(2),(2,1),(2),(5,1),(1),(1)) with leaders (3,2,2,2,5,1,1), so a(1234567) = 16.

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

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

%Y Row sums of A374757.

%Y For leaders of constant runs we have A373953.

%Y For leaders of anti-runs we have A374516.

%Y For leaders of weakly increasing runs we have A374630.

%Y For length instead of sum we have A124769.

%Y The opposite version is A374684, sum of A374683 (length A124768).

%Y The case of partitions ranked by Heinz numbers is A374706.

%Y The weak version is A374741, sum of A374740 (length A124765).

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

%Y - Length is A000120.

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

%Y - Leader is A065120.

%Y - Parts are listed by A066099.

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

%Y - Run-length transform is A333627, sum A070939.

%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 Six types of runs:

%Y - Count: A124766, A124765, A124768, A124769, A333381, A124767.

%Y - Leaders: A374629, A374740, A374683, A374757, A374515, A374251.

%Y - Rank: A375123, A375124, A375125, A375126, A375127, A373948.

%Y Cf. A106356, A188920, A189076, A233564, A238343, A272919, A333213, A373949, A374700, A374759, A374761, A374767.

%K nonn

%O 0,3

%A _Gus Wiseman_, Jul 29 2024