OFFSET
0,2
COMMENTS
The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.
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.
LINKS
EXAMPLE
the 1234567th composition in standard order is (3,2,1,2,2,1,2,5,1,1,1), with strictly decreasing runs ((3,2,1),(2),(2,1),(2),(5,1),(1),(1)), so row 1234567 is (3,2,2,2,5,1,1).
The nonnegative integers, corresponding compositions, and leaders of strictly decreasing runs begin:
0: () -> () 15: (1,1,1,1) -> (1,1,1,1)
1: (1) -> (1) 16: (5) -> (5)
2: (2) -> (2) 17: (4,1) -> (4)
3: (1,1) -> (1,1) 18: (3,2) -> (3)
4: (3) -> (3) 19: (3,1,1) -> (3,1)
5: (2,1) -> (2) 20: (2,3) -> (2,3)
6: (1,2) -> (1,2) 21: (2,2,1) -> (2,2)
7: (1,1,1) -> (1,1,1) 22: (2,1,2) -> (2,2)
8: (4) -> (4) 23: (2,1,1,1) -> (2,1,1)
9: (3,1) -> (3) 24: (1,4) -> (1,4)
10: (2,2) -> (2,2) 25: (1,3,1) -> (1,3)
11: (2,1,1) -> (2,1) 26: (1,2,2) -> (1,2,2)
12: (1,3) -> (1,3) 27: (1,2,1,1) -> (1,2,1)
13: (1,2,1) -> (1,2) 28: (1,1,3) -> (1,1,3)
14: (1,1,2) -> (1,1,2) 29: (1,1,2,1) -> (1,1,2)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[First/@Split[stc[n], Greater], {n, 0, 100}]
CROSSREFS
Row-leaders of nonempty rows are A065120.
Row-lengths are A124769.
Row-sums are A374758.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Parts are listed by A066099.
Six types of runs:
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 29 2024
STATUS
approved