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Numbers k such that the leaders of maximal anti-runs in the k-th composition in standard order (A066099) are not distinct.
6

%I #5 Aug 06 2024 20:30:21

%S 3,7,10,14,15,21,23,27,28,29,30,31,36,39,42,43,47,51,55,56,57,58,59,

%T 60,61,62,63,71,73,79,84,85,86,87,90,94,95,99,103,106,107,110,111,112,

%U 113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,135

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

%C 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 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 The sequence of terms together with the corresponding compositions begins:

%e 3: (1,1)

%e 7: (1,1,1)

%e 10: (2,2)

%e 14: (1,1,2)

%e 15: (1,1,1,1)

%e 21: (2,2,1)

%e 23: (2,1,1,1)

%e 27: (1,2,1,1)

%e 28: (1,1,3)

%e 29: (1,1,2,1)

%e 30: (1,1,1,2)

%e 31: (1,1,1,1,1)

%t stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t Select[Range[0,100],!UnsameQ@@First/@Split[stc[#],UnsameQ]&]

%Y First differs from A335466 in lacking 166, complement A335467.

%Y The complement for leaders of identical runs is A374249, counted by A274174.

%Y For leaders of identical runs we have A374253, counted by A335548.

%Y Positions of non-distinct (or non-strict) rows in A374515.

%Y The complement is A374638, counted by A374518.

%Y For identical instead of non-distinct we have A374519, counted by A374517.

%Y For identical instead of distinct we have A374520, counted by A374640.

%Y Compositions of this type are counted by A374678.

%Y Other functional neighbors are A374768, A374698, A374701, A374767.

%Y A065120 gives leaders of standard compositions.

%Y A106356 counts compositions by number of maximal anti-runs.

%Y A238279 counts compositions by number of maximal runs

%Y All of the following pertain to compositions in standard order:

%Y - Length is A000120.

%Y - Sum is A029837(n+1).

%Y - Parts are listed by A066099.

%Y - Number of adjacent equal pairs is A124762, unequal A333382.

%Y - Anti-runs are ranked by A333489, counted by A003242.

%Y - Run-length transform is A333627, sum A070939.

%Y - Run-compression transform is A373948, sum A373953, excess A373954.

%Y Six types of maximal runs:

%Y - Count: A124766, A124765, A124768, A124769, A333381, A124767.

%Y - Leaders: A374629, A374740, A374683, A374757, A374515, A374251.

%Y - Rank: A375123, A375124, A375125, A375126, A375127, A373948.

%Y Cf. A029931, A114994, A228351, A233564, A238343, A272919, A335466, A373949.

%K nonn

%O 1,1

%A _Gus Wiseman_, Aug 06 2024