OFFSET
0,6
COMMENTS
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.
LINKS
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)
EXAMPLE
The number 3310 has binary expansion 110011101110 and standard composition (1,3,1,1,2,1,1,2), with runs (1), (3), (1,1), (2), (1,1), (2), of which 4 are distinct, so a(3310) = 4.
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Length[Union[Split[stc[n]]]], {n, 0, 100}]
CROSSREFS
Counting not necessarily distinct runs gives A124767.
Using binary expansions instead of standard compositions gives A297770.
Positions of first appearances are A351015.
A005811 counts runs in binary expansion.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has distinct run-lengths.
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:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Strict compositions are A233564.
- Constant compositions are A272919.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 07 2022
STATUS
approved