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A374758 Sum of leaders of strictly decreasing runs in the n-th composition in standard order.

%I A374758 #8 Jul 31 2024 09:09:24
%S A374758 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 A374758 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 A374758 4,4,6,6,7,6,5,4,6,5,6,6,7,6,5,6,7,6,6
%N A374758 Sum of leaders of strictly decreasing runs in the n-th composition in standard order.
%C A374758 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 A374758 Gus Wiseman, <a href="/A373403/a373403.txt">Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers</a>.
%e A374758 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 A374758 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A374758 Table[Total[First/@Split[stc[n],Greater]],{n,0,100}]
%Y A374758 Row sums of A374757.
%Y A374758 For leaders of constant runs we have A373953.
%Y A374758 For leaders of anti-runs we have A374516.
%Y A374758 For leaders of weakly increasing runs we have A374630.
%Y A374758 For length instead of sum we have A124769.
%Y A374758 The opposite version is A374684, sum of A374683 (length A124768).
%Y A374758 The case of partitions ranked by Heinz numbers is A374706.
%Y A374758 The weak version is A374741, sum of A374740 (length A124765).
%Y A374758 All of the following pertain to compositions in standard order:
%Y A374758 - Length is A000120.
%Y A374758 - Sum is A029837(n+1).
%Y A374758 - Leader is A065120.
%Y A374758 - Parts are listed by A066099.
%Y A374758 - Number of adjacent equal pairs is A124762, unequal A333382.
%Y A374758 - Run-length transform is A333627, sum A070939.
%Y A374758 - Run-compression transform is A373948.
%Y A374758 - Ranks of contiguous compositions are A374249, counted by A274174.
%Y A374758 - Ranks of non-contiguous compositions are A374253, counted by A335548.
%Y A374758 Six types of runs:
%Y A374758 - Count: A124766, A124765, A124768, A124769, A333381, A124767.
%Y A374758 - Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
%Y A374758 - Rank: A375123, A375124, A375125, A375126, A375127, A373948.
%Y A374758 Cf. A106356, A188920, A189076, A233564, A238343, A272919, A333213, A373949, A374700, A374759, A374761, A374767.
%K A374758 nonn
%O A374758 0,3
%A A374758 _Gus Wiseman_, Jul 29 2024