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A351290
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Numbers k such that the k-th composition in standard order has all distinct runs.
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22
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0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 50, 51, 52, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 78
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OFFSET
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1,3
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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.
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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)
2: 10 (2)
3: 11 (1,1)
4: 100 (3)
5: 101 (2,1)
6: 110 (1,2)
7: 111 (1,1,1)
8: 1000 (4)
9: 1001 (3,1)
10: 1010 (2,2)
11: 1011 (2,1,1)
12: 1100 (1,3)
14: 1110 (1,1,2)
15: 1111 (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;
Select[Range[0, 100], UnsameQ@@Split[stc[#]]&]
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CROSSREFS
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The version for Heinz numbers and prime multiplicities is A130091.
The version for run-lengths instead of runs is A329739.
These compositions are counted by A351013.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A085207 represents concatenation of standard compositions, reverse A085208.
A351204 counts partitions where every permutation has all distinct runs.
Counting words with all distinct runs:
- A351202 = permutations of prime factors.
Selected statistics of standard compositions:
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Constant compositions are A272919.
Cf. A098859, A106356, A113835, A116608, A238279, A242882, A318928, A325545, A328592, A329745, A350952, A351015, A351201.
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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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