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A374638 Numbers k such that the leaders of anti-runs in the k-th composition in standard order (A066099) are distinct.

%I A374638 #5 Aug 02 2024 08:58:46
%S A374638 0,1,2,4,5,6,8,9,11,12,13,16,17,18,19,20,22,24,25,26,32,33,34,35,37,
%T A374638 38,40,41,44,45,46,48,49,50,52,53,54,64,65,66,67,68,69,70,72,74,75,76,
%U A374638 77,78,80,81,82,83,88,89,91,92,93,96,97,98,100,101,102,104
%N A374638 Numbers k such that the leaders of anti-runs in the k-th composition in standard order (A066099) are distinct.
%C A374638 The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.
%C A374638 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 A374638 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374638 The terms together with corresponding compositions begin:
%e A374638    0: ()
%e A374638    1: (1)
%e A374638    2: (2)
%e A374638    4: (3)
%e A374638    5: (2,1)
%e A374638    6: (1,2)
%e A374638    8: (4)
%e A374638    9: (3,1)
%e A374638   11: (2,1,1)
%e A374638   12: (1,3)
%e A374638   13: (1,2,1)
%e A374638   16: (5)
%e A374638   17: (4,1)
%e A374638   18: (3,2)
%e A374638   19: (3,1,1)
%e A374638   20: (2,3)
%e A374638   22: (2,1,2)
%e A374638   24: (1,4)
%e A374638   25: (1,3,1)
%e A374638   26: (1,2,2)
%t A374638 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A374638 Select[Range[0,100],UnsameQ@@First/@Split[stc[#],UnsameQ]&]
%Y A374638 Positions of distinct (strict) rows in A374515.
%Y A374638 Compositions of this type are counted by A374518.
%Y A374638 For identical instead of distinct we have A374519, counted by A374517.
%Y A374638 The complement is A374639.
%Y A374638 Other types of runs (instead of anti-):
%Y A374638 - For identical runs we have A374249, counted by A274174.
%Y A374638 - For weakly increasing runs we have A374768, counted by A374632.
%Y A374638 - For strictly increasing runs we have A374698, counted by A374687.
%Y A374638 - For weakly decreasing runs we have A374701, counted by A374743.
%Y A374638 - For strictly decreasing runs we have A374767, counted by A374761.
%Y A374638 A065120 gives leaders of standard compositions.
%Y A374638 A106356 counts compositions by number of maximal anti-runs.
%Y A374638 A238279 counts compositions by number of maximal runs
%Y A374638 A238424 counts partitions whose first differences are an anti-run.
%Y A374638 All of the following pertain to compositions in standard order:
%Y A374638 - Length is A000120.
%Y A374638 - Sum is A029837(n+1).
%Y A374638 - Parts are listed by A066099.
%Y A374638 - Number of adjacent equal pairs is A124762, unequal A333382.
%Y A374638 - Anti-runs are ranked by A333489, counted by A003242.
%Y A374638 - Run-length transform is A333627, sum A070939.
%Y A374638 - Run-compression transform is A373948, sum A373953, excess A373954.
%Y A374638 Six types of maximal runs:
%Y A374638 - Count: A124766, A124765, A124768, A124769, A333381, A124767.
%Y A374638 - Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
%Y A374638 - Rank: A375123, A375124, A375125, A375126, A375127, A373948.
%Y A374638 Cf. A029931, A114994, A228351, A233564, A238343, A272919, A335467, A373949.
%K A374638 nonn
%O A374638 1,3
%A A374638 _Gus Wiseman_, Aug 01 2024