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A374630 Sum of leaders of weakly increasing runs in the n-th composition in standard order.

%I A374630 #6 Jul 22 2024 21:11:20
%S A374630 0,1,2,1,3,3,1,1,4,4,2,3,1,2,1,1,5,5,5,4,2,3,3,3,1,2,1,2,1,2,1,1,6,6,
%T A374630 6,5,3,6,4,4,2,3,2,3,3,4,3,3,1,2,3,2,1,2,2,2,1,2,1,2,1,2,1,1,7,7,7,6,
%U A374630 7,7,5,5,3,4,5,6,4,5,4,4,2,3,4,3,2,3,3
%N A374630 Sum of leaders of weakly increasing runs in the n-th composition in standard order.
%C A374630 The leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.
%C A374630 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 A374630 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374630 The maximal weakly increasing subsequences of the 1234567th composition in standard order are ((3),(2),(1,2,2),(1,2,5),(1,1,1)), so a(1234567) = 8.
%t A374630 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A374630 Table[Total[First/@Split[stc[n],LessEqual]],{n,0,100}]
%Y A374630 For length instead of sum we have A124766.
%Y A374630 For leaders of constant runs we have A373953, excess A373954.
%Y A374630 For leaders of anti-runs we have A374516.
%Y A374630 Row-sums of A374629.
%Y A374630 Counting compositions by this statistic gives A374637.
%Y A374630 For leaders of strictly increasing runs we have A374684.
%Y A374630 For leaders of weakly decreasing runs we have A374741.
%Y A374630 For leaders of strictly decreasing runs we have A374758
%Y A374630 A011782 counts compositions.
%Y A374630 A238130, A238279, A333755 count compositions by number of runs.
%Y A374630 A335456 counts patterns matched by compositions.
%Y A374630 A373949 counts compositions by run-compressed sum, opposite A373951.
%Y A374630 All of the following pertain to compositions in standard order:
%Y A374630 - Ones are counted by A000120.
%Y A374630 - Sum is A029837 (or sometimes A070939).
%Y A374630 - Listed by A066099.
%Y A374630 - Length is A070939.
%Y A374630 - Number of adjacent equal pairs is A124762, unequal A333382.
%Y A374630 - Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
%Y A374630 - Ranks of strict compositions are A233564, counted by A032020.
%Y A374630 - Constant compositions are ranked by A272919.
%Y A374630 - Ranks of anti-run compositions are A333489, counted by A003242.
%Y A374630 - Run-length transform is A333627.
%Y A374630 - Run-compression transform is A373948.
%Y A374630 Cf. A065120, A106356, A188920, A189076, A238343, A333175, A333213, A374631, A374633, A374635.
%K A374630 nonn
%O A374630 0,3
%A A374630 _Gus Wiseman_, Jul 20 2024