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A374683
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Irregular triangle read by rows where row n lists the leaders of strictly increasing runs in the n-th composition in standard order.
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28
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1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 4, 3, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 4, 1, 3, 2, 3, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 5, 1, 4, 2, 4, 1, 1, 3, 3, 3, 2, 1, 3, 1, 3, 1, 1, 1
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OFFSET
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0,2
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COMMENTS
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The leaders of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.
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.
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LINKS
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EXAMPLE
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The maximal strictly increasing subsequences of the 1234567th composition in standard order are ((3),(2),(1,2),(2),(1,2,5),(1),(1),(1)), so row 1234567 is (3,2,1,2,1,1,1,1).
The nonnegative integers, corresponding compositions, and leaders of strictly increasing runs begin:
0: () -> () 15: (1,1,1,1) -> (1,1,1,1)
1: (1) -> (1) 16: (5) -> (5)
2: (2) -> (2) 17: (4,1) -> (4,1)
3: (1,1) -> (1,1) 18: (3,2) -> (3,2)
4: (3) -> (3) 19: (3,1,1) -> (3,1,1)
5: (2,1) -> (2,1) 20: (2,3) -> (2)
6: (1,2) -> (1) 21: (2,2,1) -> (2,2,1)
7: (1,1,1) -> (1,1,1) 22: (2,1,2) -> (2,1)
8: (4) -> (4) 23: (2,1,1,1) -> (2,1,1,1)
9: (3,1) -> (3,1) 24: (1,4) -> (1)
10: (2,2) -> (2,2) 25: (1,3,1) -> (1,1)
11: (2,1,1) -> (2,1,1) 26: (1,2,2) -> (1,2)
12: (1,3) -> (1) 27: (1,2,1,1) -> (1,1,1)
13: (1,2,1) -> (1,1) 28: (1,1,3) -> (1,1)
14: (1,1,2) -> (1,1) 29: (1,1,2,1) -> (1,1,1)
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MATHEMATICA
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stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[First/@Split[stc[n], Less], {n, 0, 100}]
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CROSSREFS
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All of the following pertain to compositions in standard order:
- Ranks of non-contiguous compositions are A374253, counted by A335548.
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KEYWORD
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nonn,tabf,new
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AUTHOR
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STATUS
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approved
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