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A374744 Numbers k such that the leaders of weakly decreasing runs in the k-th composition in standard order (A066099) are identical.

%I A374744 #6 Jul 26 2024 08:58:34
%S A374744 0,1,2,3,4,5,7,8,9,10,11,15,16,17,18,19,21,22,23,31,32,33,34,35,36,37,
%T A374744 39,42,43,45,46,47,63,64,65,66,67,68,69,71,73,74,75,76,79,85,86,87,90,
%U A374744 91,93,94,95,127,128,129,130,131,132,133,135,136,137,138
%N A374744 Numbers k such that the leaders of weakly decreasing runs in the k-th composition in standard order (A066099) are identical.
%C A374744 The leaders of weakly decreasing runs in a sequence are obtained by splitting into maximal weakly decreasing subsequences and taking the first term of each.
%C A374744 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 A374744 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374744 The terms together with the corresponding compositions begin:
%e A374744    0: ()
%e A374744    1: (1)
%e A374744    2: (2)
%e A374744    3: (1,1)
%e A374744    4: (3)
%e A374744    5: (2,1)
%e A374744    7: (1,1,1)
%e A374744    8: (4)
%e A374744    9: (3,1)
%e A374744   10: (2,2)
%e A374744   11: (2,1,1)
%e A374744   15: (1,1,1,1)
%e A374744   16: (5)
%e A374744   17: (4,1)
%e A374744   18: (3,2)
%e A374744   19: (3,1,1)
%e A374744   21: (2,2,1)
%e A374744   22: (2,1,2)
%e A374744   23: (2,1,1,1)
%e A374744   31: (1,1,1,1,1)
%t A374744 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A374744 Select[Range[0,100],SameQ@@First/@Split[stc[#],GreaterEqual]&]
%Y A374744 Other types of runs and their counts: A272919 (A000005), A374519 (A374517), A374685 (A374686), A374759 (A374760).
%Y A374744 The opposite is A374633, counted by A374631.
%Y A374744 For distinct (instead of identical) leaders we have A374701, count A374743.
%Y A374744 Positions of constant rows in A374740, opposite A374629, cf. A374630.
%Y A374744 Compositions of this type are counted by A374742.
%Y A374744 A011782 counts compositions.
%Y A374744 A238130, A238279, A333755 count compositions by number of runs.
%Y A374744 A374748 counts compositions by sum of leaders of weakly decreasing runs.
%Y A374744 All of the following pertain to compositions in standard order:
%Y A374744 - Length is A000120.
%Y A374744 - Sum is A029837(n+1) (or sometimes A070939).
%Y A374744 - Parts are listed by A066099.
%Y A374744 - Adjacent equal pairs are counted by A124762, unequal A333382.
%Y A374744 - Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
%Y A374744 - Ranks of anti-run compositions are A333489, counted by A003242.
%Y A374744 - Run-length transform is A333627.
%Y A374744 - Run-compression transform is A373948, sum A373953, excess A373954.
%Y A374744 - Ranks of contiguous compositions are A374249, counted by A274174.
%Y A374744 Cf. A065120, A106356, A238343, A333175, A333213, A335456, A373949, A374635, A374634, A374768.
%K A374744 nonn
%O A374744 1,3
%A A374744 _Gus Wiseman_, Jul 24 2024