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Irregular triangle read by rows where row n lists the leaders of weakly decreasing runs in the n-th composition in standard order.
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%I #8 Jul 26 2024 08:58:38

%S 1,2,1,3,2,1,2,1,4,3,2,2,1,3,1,2,1,2,1,5,4,3,3,2,3,2,2,2,2,1,4,1,3,1,

%T 2,1,2,1,3,1,2,1,2,1,6,5,4,4,3,3,3,2,3,2,4,2,3,2,2,2,3,2,2,2,2,2,1,5,

%U 1,4,1,3,1,3,1,2,3,1,2,1,2,2,1,2,1,4

%N Irregular triangle read by rows where row n lists the leaders of weakly decreasing runs in the n-th composition in standard order.

%C The leaders of weakly decreasing runs in a sequence are obtained by splitting it 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 maximal weakly decreasing subsequences of the 1234567th composition in standard order are ((3,2,1),(2,2,1),(2),(5,1,1,1)), so row 1234567 is (3,2,2,5).

%e The nonnegative integers, corresponding compositions, and leaders of weakly decreasing runs begin:

%e 0: () -> () 15: (1,1,1,1) -> (1)

%e 1: (1) -> (1) 16: (5) -> (5)

%e 2: (2) -> (2) 17: (4,1) -> (4)

%e 3: (1,1) -> (1) 18: (3,2) -> (3)

%e 4: (3) -> (3) 19: (3,1,1) -> (3)

%e 5: (2,1) -> (2) 20: (2,3) -> (2,3)

%e 6: (1,2) -> (1,2) 21: (2,2,1) -> (2)

%e 7: (1,1,1) -> (1) 22: (2,1,2) -> (2,2)

%e 8: (4) -> (4) 23: (2,1,1,1) -> (2)

%e 9: (3,1) -> (3) 24: (1,4) -> (1,4)

%e 10: (2,2) -> (2) 25: (1,3,1) -> (1,3)

%e 11: (2,1,1) -> (2) 26: (1,2,2) -> (1,2)

%e 12: (1,3) -> (1,3) 27: (1,2,1,1) -> (1,2)

%e 13: (1,2,1) -> (1,2) 28: (1,1,3) -> (1,3)

%e 14: (1,1,2) -> (1,2) 29: (1,1,2,1) -> (1,2)

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

%t Table[First/@Split[stc[n],GreaterEqual],{n,0,100}]

%Y Row-leaders are A065120.

%Y Row-lengths are A124765.

%Y Other types of runs are A374251, A374515, A374683, A374757.

%Y The opposite is A374629.

%Y Positions of distinct (strict) rows are A374701, counted by A374743.

%Y Row-sums are A374741, opposite A374630.

%Y Positions of identical rows are A374744, counted by A374742.

%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 - Number of adjacent equal pairs is 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, sum A070939.

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

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

%Y - Ranks of non-contiguous compositions are A374253, counted by A335548.

%Y Cf. A106356, A188920, A189076, A238343, A272919, A333213, A373949, A374634, A374635, A374637.

%K nonn,tabf

%O 0,2

%A _Gus Wiseman_, Jul 24 2024