OFFSET
0,2
COMMENTS
The leaders of weakly decreasing runs in a sequence are obtained by splitting it into maximal weakly 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 maximal weakly decreasing subsequences of the 1234567th composition in standard order are ((3,2,1),(2,2,1),(2),(5,1,1,1)), so row 1234567 is (3,2,2,5).
The nonnegative integers, corresponding compositions, and leaders of weakly decreasing runs begin:
0: () -> () 15: (1,1,1,1) -> (1)
1: (1) -> (1) 16: (5) -> (5)
2: (2) -> (2) 17: (4,1) -> (4)
3: (1,1) -> (1) 18: (3,2) -> (3)
4: (3) -> (3) 19: (3,1,1) -> (3)
5: (2,1) -> (2) 20: (2,3) -> (2,3)
6: (1,2) -> (1,2) 21: (2,2,1) -> (2)
7: (1,1,1) -> (1) 22: (2,1,2) -> (2,2)
8: (4) -> (4) 23: (2,1,1,1) -> (2)
9: (3,1) -> (3) 24: (1,4) -> (1,4)
10: (2,2) -> (2) 25: (1,3,1) -> (1,3)
11: (2,1,1) -> (2) 26: (1,2,2) -> (1,2)
12: (1,3) -> (1,3) 27: (1,2,1,1) -> (1,2)
13: (1,2,1) -> (1,2) 28: (1,1,3) -> (1,3)
14: (1,1,2) -> (1,2) 29: (1,1,2,1) -> (1,2)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[First/@Split[stc[n], GreaterEqual], {n, 0, 100}]
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Jul 24 2024
STATUS
approved