OFFSET
0,3
COMMENTS
A composition of n is a finite sequence of positive integers summing to n. 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.
EXAMPLE
The standard compositions and their run-lengths:
0: () -> ()
1: (1) -> (1)
2: (2) -> (1)
3: (1,1) -> (2)
4: (3) -> (1)
5: (2,1) -> (1,1)
6: (1,2) -> (1,1)
7: (1,1,1) -> (3)
8: (4) -> (1)
9: (3,1) -> (1,1)
10: (2,2) -> (2)
11: (2,1,1) -> (1,2)
12: (1,3) -> (1,1)
13: (1,2,1) -> (1,1,1)
14: (1,1,2) -> (2,1)
15: (1,1,1,1) -> (4)
16: (5) -> (1)
17: (4,1) -> (1,1)
18: (3,2) -> (1,1)
19: (3,1,1) -> (1,2)
For example, the 119th composition is (1,1,2,1,1,1), so row 119 is (2,1,3).
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Length/@Split[stc[n]], {n, 0, 30}]
CROSSREFS
Row sums are A000120.
Row lengths are A124767.
Row k is the A333627(k)-th standard composition.
A triangle counting compositions by runs-resistance is A329744.
All of the following pertain to compositions in standard order (A066099):
- Partial sums from the right are A048793.
- Sum is A070939.
- Adjacent equal pairs are counted by A124762.
- Strict compositions are A233564.
- Partial sums from the left are A272020.
- Constant compositions are A272919.
- Normal compositions are A333217.
- Heinz number is A333219.
- Runs-resistance is A333628.
- First appearances of run-resistances are A333629.
- Combinatory separations are A334030.
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Apr 10 2020
STATUS
approved