A374768 - OEIS

%I #13 Jul 22 2024 21:11:11

%S 0,1,2,3,4,5,6,7,8,9,10,11,12,14,15,16,17,18,19,20,21,22,23,24,26,28,

%T 30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,46,47,48,50,52,56,58,60,

%U 62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,78,79,80,81

%N Numbers k such that the leaders of weakly increasing runs in the k-th composition in standard order (A066099) are distinct.

%C First differs from A335467 in having 166, corresponding to the composition (2,3,1,2).

%C The leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences 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.

%H Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.

%e The 4444th composition in standard order is (4,2,2,1,1,3), with weakly increasing runs ((4),(2,2),(1,1,3)), with leaders (4,2,1), so 4444 is in the sequence.

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

%t Select[Range[0,166],UnsameQ@@First/@Split[stc[#],LessEqual]&]

%Y These are the positions of strict rows in A374629 (which has sums A374630).

%Y Compositions of this type are counted by A374632, increasing A374634.

%Y Identical instead of distinct leaders are A374633, counted by A374631.

%Y For leaders of anti-runs we have A374638, counted by A374518.

%Y For leaders of strictly increasing runs we have A374698, counted by A374687.

%Y For leaders of weakly decreasing runs we have A374701, counted by A374743.

%Y For leaders of strictly decreasing runs we have A374767, counted by A374761.

%Y A011782 counts compositions.

%Y A238130, A238279, A333755 count compositions by number of runs.

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

%Y - Ones are counted by A000120.

%Y - Sum is A029837 (or sometimes A070939).

%Y - Parts are listed by A066099.

%Y - Length is A070939.

%Y - Adjacent equal pairs are counted by A124762, unequal A333382.

%Y - Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.

%Y - Ranks of strict compositions are A233564.

%Y - Ranks of constant compositions are A272919.

%Y - Ranks of anti-run compositions are A333489, counted by A003242.

%Y - Run-length transform is A333627.

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

%Y Cf. A065120, A188920, A189076, A238343, A333213, A335449, A373949, A274174, A374249, A374635, A374637.

%K nonn

%O 1,3

%A _Gus Wiseman_, Jul 19 2024