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A374639 Numbers k such that the leaders of maximal anti-runs in the k-th composition in standard order (A066099) are not distinct.

%I A374639 #5 Aug 06 2024 20:30:21
%S A374639 3,7,10,14,15,21,23,27,28,29,30,31,36,39,42,43,47,51,55,56,57,58,59,
%T A374639 60,61,62,63,71,73,79,84,85,86,87,90,94,95,99,103,106,107,110,111,112,
%U A374639 113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,135
%N A374639 Numbers k such that the leaders of maximal anti-runs in the k-th composition in standard order (A066099) are not distinct.
%C A374639 The leaders of maximal 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 A374639 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.
%e A374639 The sequence of terms together with the corresponding compositions begins:
%e A374639    3: (1,1)
%e A374639    7: (1,1,1)
%e A374639   10: (2,2)
%e A374639   14: (1,1,2)
%e A374639   15: (1,1,1,1)
%e A374639   21: (2,2,1)
%e A374639   23: (2,1,1,1)
%e A374639   27: (1,2,1,1)
%e A374639   28: (1,1,3)
%e A374639   29: (1,1,2,1)
%e A374639   30: (1,1,1,2)
%e A374639   31: (1,1,1,1,1)
%t A374639 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A374639 Select[Range[0,100],!UnsameQ@@First/@Split[stc[#],UnsameQ]&]
%Y A374639 First differs from A335466 in lacking 166, complement A335467.
%Y A374639 The complement for leaders of identical runs is A374249, counted by A274174.
%Y A374639 For leaders of identical runs we have A374253, counted by A335548.
%Y A374639 Positions of non-distinct (or non-strict) rows in A374515.
%Y A374639 The complement is A374638, counted by A374518.
%Y A374639 For identical instead of non-distinct we have A374519, counted by A374517.
%Y A374639 For identical instead of distinct we have A374520, counted by A374640.
%Y A374639 Compositions of this type are counted by A374678.
%Y A374639 Other functional neighbors are A374768, A374698, A374701, A374767.
%Y A374639 A065120 gives leaders of standard compositions.
%Y A374639 A106356 counts compositions by number of maximal anti-runs.
%Y A374639 A238279 counts compositions by number of maximal runs
%Y A374639 All of the following pertain to compositions in standard order:
%Y A374639 - Length is A000120.
%Y A374639 - Sum is A029837(n+1).
%Y A374639 - Parts are listed by A066099.
%Y A374639 - Number of adjacent equal pairs is A124762, unequal A333382.
%Y A374639 - Anti-runs are ranked by A333489, counted by A003242.
%Y A374639 - Run-length transform is A333627, sum A070939.
%Y A374639 - Run-compression transform is A373948, sum A373953, excess A373954.
%Y A374639 Six types of maximal runs:
%Y A374639 - Count: A124766, A124765, A124768, A124769, A333381, A124767.
%Y A374639 - Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
%Y A374639 - Rank: A375123, A375124, A375125, A375126, A375127, A373948.
%Y A374639 Cf. A029931, A114994, A228351, A233564, A238343, A272919, A335466, A373949.
%K A374639 nonn
%O A374639 1,1
%A A374639 _Gus Wiseman_, Aug 06 2024