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.
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)
7: 111 (1,1,1)
8: 1000 (4)
10: 1010 (2,2)
11: 1011 (2,1,1)
14: 1110 (1,1,2)
15: 1111 (1,1,1,1)
16: 10000 (5)
19: 10011 (3,1,1)
21: 10101 (2,2,1)
23: 10111 (2,1,1,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], UnsameQ@@Length/@Split[stc[#]]&]
CROSSREFS
The version using binary expansions is A044813.
The version for Heinz numbers and prime multiplicities is A130091.
A011782 counts integer compositions.
A351204 counts partitions where every permutation has all distinct runs.
Counting words with all distinct run-lengths:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of distinct parts is A334028.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 24 2022
STATUS
approved