OFFSET
1,3
COMMENTS
The leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing 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 increasing subsequences of the 26165th composition in standard order are ((1,3),(1,4),(1,2,2),(1)), with leaders (1,1,1,1), so 26165 is in the sequence.
The sequence together with the corresponding compositions begins:
0: ()
1: (1)
2: (2)
3: (1,1)
4: (3)
6: (1,2)
7: (1,1,1)
8: (4)
10: (2,2)
12: (1,3)
13: (1,2,1)
14: (1,1,2)
15: (1,1,1,1)
16: (5)
20: (2,3)
24: (1,4)
25: (1,3,1)
26: (1,2,2)
27: (1,2,1,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], SameQ@@First/@Split[stc[#], LessEqual]&]
CROSSREFS
For strictly decreasing leaders we appear to have A188920.
For weakly decreasing leaders we appear to have A189076.
Other types of runs: A272919 (counted by A000005), A374519 (counted by A374517), A374685 (counted by A374686), A374744 (counted by A374742), A374759 (counted by A374760).
Compositions of this type are counted by A374631.
For strictly increasing leaders see A374634.
A011782 counts compositions.
A374637 counts compositions by sum of leaders of weakly increasing runs.
All of the following pertain to compositions in standard order:
- Ones are counted by A000120.
- Parts are listed by A066099.
- Length is A070939.
- Run-length transform is A333627.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 21 2024
STATUS
approved