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A354579
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Number of distinct lengths of runs in the n-th composition in standard order.
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3
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0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 2
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OFFSET
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0,12
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COMMENTS
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Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with lengths (2,3,1,2).
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 positions of first appearances together with the corresponding compositions begin:
1: (1)
11: (2,1,1)
119: (1,1,2,1,1,1)
5615: (2,2,1,1,1,2,1,1,1,1)
251871: (1,1,1,2,2,1,1,1,1,2,1,1,1,1,1)
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MATHEMATICA
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stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Length[Union[Length/@Split[stc[n]]]], {n, 0, 100}]
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CROSSREFS
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Standard compositions are listed by A066099.
The version for partitions is A071625.
Positions of first appearances are A354906.
A005811 counts runs in binary expansion.
A333627 ranks the run-lengths of standard compositions.
A351596 ranks compositions with distinct run-lengths, counted by A329739.
A353847 ranks the run-sums of standard compositions.
A353852 ranks compositions with distinct run-sums, counted by A353850.
A353860 counts collapsible compositions.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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