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A374519 Numbers k such that the leaders of anti-runs in the k-th composition in standard order (A066099) are identical.

%I A374519 #5 Aug 02 2024 08:58:29
%S A374519 0,1,2,3,4,5,6,7,8,9,10,12,13,14,15,16,17,18,20,21,22,24,25,27,28,29,
%T A374519 30,31,32,33,34,36,37,38,40,41,42,44,45,48,49,50,51,52,54,55,56,57,59,
%U A374519 60,61,62,63,64,65,66,68,69,70,72,73,76,77,80,81,82,84,85
%N A374519 Numbers k such that the leaders of anti-runs in the k-th composition in standard order (A066099) are identical.
%C A374519 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 A374519 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 A374519 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374519 The 346th composition in standard order is (2,2,1,2,2), with anti-runs ((2),(2,1,2),(2)), with leaders (2,2,2), so 346 is in the sequence.
%t A374519 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A374519 Select[Range[0,100],SameQ@@First/@Split[stc[#],UnsameQ]&]
%Y A374519 Positions of constant rows in A374515.
%Y A374519 Compositions of this type are counted by A374517.
%Y A374519 The complement is A374520.
%Y A374519 For distinct instead of identical leaders we have A374638, counted by A374518.
%Y A374519 Other types of runs (instead of anti-):
%Y A374519 - For identical runs we have A272919, counted by A000005.
%Y A374519 - For weakly increasing runs we have A374633, counted by A374631.
%Y A374519 - For strictly increasing runs we have A374685, counted by A374686.
%Y A374519 - For weakly decreasing runs we have A374744, counted by A374742.
%Y A374519 - For strictly decreasing runs we have A374759, counted by A374760.
%Y A374519 A065120 gives leaders of standard compositions.
%Y A374519 A106356 counts compositions by number of maximal anti-runs.
%Y A374519 A238279 counts compositions by number of maximal runs.
%Y A374519 A238424 counts partitions whose first differences are an anti-run.
%Y A374519 All of the following pertain to compositions in standard order:
%Y A374519 - Length is A000120.
%Y A374519 - Sum is A029837(n+1).
%Y A374519 - Parts are listed by A066099.
%Y A374519 - Number of adjacent equal pairs is A124762, unequal A333382.
%Y A374519 - Anti-runs are ranked by A333489, counted by A003242.
%Y A374519 - Run-length transform is A333627, sum A070939.
%Y A374519 - Run-compression transform is A373948, sum A373953, excess A373954.
%Y A374519 - Ranks of contiguous compositions are A374249, counted by A274174.
%Y A374519 Six types of maximal runs:
%Y A374519 - Count: A124766, A124765, A124768, A124769, A333381, A124767.
%Y A374519 - Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
%Y A374519 - Rank: A375123, A375124, A375125, A375126, A375127, A373948.
%Y A374519 Cf. A029931, A114994, A228351, A233564, A238343, A373949.
%K A374519 nonn
%O A374519 1,3
%A A374519 _Gus Wiseman_, Aug 01 2024