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Numbers k such that the leaders of weakly decreasing runs in the k-th composition in standard order (A066099) are identical.
17

%I #6 Jul 26 2024 08:58:34

%S 0,1,2,3,4,5,7,8,9,10,11,15,16,17,18,19,21,22,23,31,32,33,34,35,36,37,

%T 39,42,43,45,46,47,63,64,65,66,67,68,69,71,73,74,75,76,79,85,86,87,90,

%U 91,93,94,95,127,128,129,130,131,132,133,135,136,137,138

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

%C The leaders of weakly decreasing runs in a sequence are obtained by splitting into maximal weakly decreasing 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 terms together with the corresponding compositions begin:

%e 0: ()

%e 1: (1)

%e 2: (2)

%e 3: (1,1)

%e 4: (3)

%e 5: (2,1)

%e 7: (1,1,1)

%e 8: (4)

%e 9: (3,1)

%e 10: (2,2)

%e 11: (2,1,1)

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

%e 16: (5)

%e 17: (4,1)

%e 18: (3,2)

%e 19: (3,1,1)

%e 21: (2,2,1)

%e 22: (2,1,2)

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

%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],SameQ@@First/@Split[stc[#],GreaterEqual]&]

%Y Other types of runs and their counts: A272919 (A000005), A374519 (A374517), A374685 (A374686), A374759 (A374760).

%Y The opposite is A374633, counted by A374631.

%Y For distinct (instead of identical) leaders we have A374701, count A374743.

%Y Positions of constant rows in A374740, opposite A374629, cf. A374630.

%Y Compositions of this type are counted by A374742.

%Y A011782 counts compositions.

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

%Y A374748 counts compositions by sum of leaders of weakly decreasing runs.

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

%Y - Length is A000120.

%Y - Sum is A029837(n+1) (or sometimes A070939).

%Y - Parts are listed by A066099.

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

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

%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 - Ranks of contiguous compositions are A374249, counted by A274174.

%Y Cf. A065120, A106356, A238343, A333175, A333213, A335456, A373949, A374635, A374634, A374768.

%K nonn

%O 1,3

%A _Gus Wiseman_, Jul 24 2024