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A374759 Numbers k such that the leaders of strictly decreasing runs in the k-th composition in standard order are identical.

%I A374759 #6 Jul 31 2024 09:09:34
%S A374759 0,1,2,3,4,5,7,8,9,10,15,16,17,18,21,22,31,32,33,34,36,37,42,45,63,64,
%T A374759 65,66,68,69,73,76,85,86,90,127,128,129,130,132,133,136,137,146,148,
%U A374759 153,170,173,181,182
%N A374759 Numbers k such that the leaders of strictly decreasing runs in the k-th composition in standard order are identical.
%C A374759 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 A374759 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374759 The 18789th composition in standard order is (3,3,2,1,3,2,1), with strictly decreasing runs ((3),(3,2,1),(3,2,1)), with leaders (3,3,3), so 18789 is in the sequence.
%e A374759 The terms together with the corresponding compositions begin:
%e A374759    0: ()
%e A374759    1: (1)
%e A374759    2: (2)
%e A374759    3: (1,1)
%e A374759    4: (3)
%e A374759    5: (2,1)
%e A374759    7: (1,1,1)
%e A374759    8: (4)
%e A374759    9: (3,1)
%e A374759   10: (2,2)
%e A374759   15: (1,1,1,1)
%e A374759   16: (5)
%e A374759   17: (4,1)
%e A374759   18: (3,2)
%e A374759   21: (2,2,1)
%e A374759   22: (2,1,2)
%e A374759   31: (1,1,1,1,1)
%e A374759   32: (6)
%e A374759   33: (5,1)
%e A374759   34: (4,2)
%e A374759   36: (3,3)
%e A374759   37: (3,2,1)
%t A374759 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A374759 Select[Range[0,100],SameQ@@First/@Split[stc[#],Greater]&]
%Y A374759 For leaders of anti-runs we have A374519 (counted by A374517).
%Y A374759 For leaders of weakly increasing runs we have A374633, counted by A374631.
%Y A374759 The opposite version is A374685 (counted by A374686).
%Y A374759 The weak version is A374744.
%Y A374759 Compositions of this type are counted by A374760.
%Y A374759 For distinct instead of identical runs we have A374767 (counted by A374761).
%Y A374759 All of the following pertain to compositions in standard order:
%Y A374759 - Length is A000120.
%Y A374759 - Sum is A029837(n+1).
%Y A374759 - Parts are listed by A066099.
%Y A374759 - Number of adjacent equal pairs is A124762, unequal A333382.
%Y A374759 - Run-length transform is A333627, sum A070939.
%Y A374759 - Run-compression transform is A373948, sum A373953, excess A373954.
%Y A374759 - Ranks of contiguous compositions are A374249, counted by A274174.
%Y A374759 - Ranks of non-contiguous compositions are A374253, counted by A335548.
%Y A374759 Six types of runs:
%Y A374759 - Count: A124766, A124765, A124768, A124769, A333381, A124767.
%Y A374759 - Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
%Y A374759 - Rank: A375123, A375124, A375125, A375126, A375127, A373948.
%Y A374759 Cf. A000961, A065120, A106356, A188920, A189076, A238343, A272919, A333213, A374698, A374706, A374758, A375128.
%K A374759 nonn
%O A374759 1,3
%A A374759 _Gus Wiseman_, Jul 29 2024