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A374741 Sum of leaders of weakly decreasing runs in the n-th composition in standard order.

%I A374741 #7 Jul 26 2024 08:58:29
%S A374741 0,1,2,1,3,2,3,1,4,3,2,2,4,3,3,1,5,4,3,3,5,2,4,2,5,4,3,3,4,3,3,1,6,5,
%T A374741 4,4,3,3,5,3,6,5,2,2,5,4,4,2,6,5,4,4,6,3,5,3,5,4,3,3,4,3,3,1,7,6,5,5,
%U A374741 4,4,6,4,7,3,3,3,6,5,5,3,7,6,5,5,5,2,4
%N A374741 Sum of leaders of weakly decreasing runs in the n-th composition in standard order.
%C A374741 The leaders of weakly decreasing runs in a sequence are obtained by splitting it into maximal weakly decreasing subsequences and taking the first term of each.
%C A374741 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 A374741 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374741 The maximal weakly decreasing subsequences of the 1234567th composition in standard order are ((3,2,1),(2,2,1),(2),(5,1,1,1)), so a(1234567) is 3+2+2+5 = 12.
%t A374741 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A374741 Table[Total[First/@Split[stc[n],GreaterEqual]],{n,0,100}]
%Y A374741 For length instead of sum we have A124765.
%Y A374741 Other types of runs are A373953, A374516, A374684, A374758.
%Y A374741 The opposite is A374630.
%Y A374741 Row-sums of A374740, opposite A374629.
%Y A374741 Counting compositions by this statistic gives A374748, opposite A374637.
%Y A374741 A373949 counts compositions by run-compressed sum.
%Y A374741 All of the following pertain to compositions in standard order:
%Y A374741 - Length is A000120.
%Y A374741 - Sum is A029837(n+1) (or sometimes A070939).
%Y A374741 - Parts are listed by A066099.
%Y A374741 - Number of adjacent equal pairs is A124762, unequal A333382.
%Y A374741 - Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
%Y A374741 - Ranks of strict compositions are A233564, counted by A032020.
%Y A374741 - Constant compositions are ranked by A272919.
%Y A374741 - Ranks of anti-run compositions are A333489, counted by A003242.
%Y A374741 - Run-length transform is A333627.
%Y A374741 - Run-compression transform is A373948.
%Y A374741 Cf. A065120, A106356, A188920, A189076, A238343, A333175, A333213, A373949, A373954, A374631, A374633, A374635, A374701, A374742.
%K A374741 nonn
%O A374741 0,3
%A A374741 _Gus Wiseman_, Jul 24 2024