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A353852
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Numbers k such that the k-th composition in standard order (row k of A066099) has all distinct run-sums.
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37
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0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 17, 18, 19, 20, 21, 23, 24, 26, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 47, 48, 50, 51, 52, 55, 56, 57, 58, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 79, 80, 81, 84, 85, 86, 87, 88
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OFFSET
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0,3
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COMMENTS
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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.
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 sums (4,3,3,4).
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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)
12: 1100 (1,3)
15: 1111 (1,1,1,1)
16: 10000 (5)
17: 10001 (4,1)
18: 10010 (3,2)
19: 10011 (3,1,1)
20: 10100 (2,3)
21: 10101 (2,2,1)
23: 10111 (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;
Select[Range[0, 100], UnsameQ@@Total/@Split[stc[#]]&]
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CROSSREFS
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The version for runs in binary expansion is A175413.
The version for parts instead of run-sums is A233564, counted A032020.
The version for run-lengths instead of run-sums is A351596, counted A329739.
The version for runs instead of run-sums is A351290, counted by A351013.
These compositions are counted by A353850.
The weak version (rucksack compositions) is A354581, counted by A354580.
A005811 counts runs in binary expansion.
A242882 counts composition with distinct multiplicities, partitions A098859.
A304442 counts partitions with all equal run-sums.
A351014 counts distinct runs in standard compositions, firsts A351015.
Cf. A044813, A238279, A333755, A351016, A351017, A353832, A353847, A353849, A353860, A353863, A353932.
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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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