OFFSET
1,3
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 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)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], UnsameQ@@Split[stc[#]]&]
CROSSREFS
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.
The complement is A351291.
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 10 2022
STATUS
approved