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A351015
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Smallest k such that the k-th composition in standard order has n distinct runs.
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28
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OFFSET
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0,3
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COMMENTS
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The n-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 n, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
It would be very interesting to have a formula or general construction for a(n). - Gus Wiseman, Feb 12 2022
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LINKS
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EXAMPLE
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The terms together with their binary expansions and corresponding compositions begin:
0: 0 ()
1: 1 (1)
5: 101 (2,1)
27: 11011 (1,2,1,1)
155: 10011011 (3,1,2,1,1)
1655: 11001110111 (1,3,1,1,2,1,1,1)
18039: 100011001110111 (4,1,3,1,1,2,1,1,1)
281975: 1000100110101110111 (4,3,1,2,2,1,1,2,1,1,1)
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MATHEMATICA
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stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
s=Table[Length[Union[Split[stc[n]]]], {n, 0, 1000}];
Table[Position[s, k][[1, 1]]-1, {k, Union[s]}]
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CROSSREFS
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The version for Heinz numbers and prime multiplicities is A006939.
Counting not necessarily distinct runs gives A113835 (up to zero).
Using binary expansions instead of standard compositions gives A350952.
These are the positions of first appearances in A351014.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A085207 represents concatenation of standard compositions, reverse A085208.
Counting words with all distinct runs:
Selected statistics of standard compositions (A066099, reverse A228351):
- Number of distinct parts is A334028.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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