OFFSET
1,3
COMMENTS
The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) 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 terms together with corresponding compositions begin:
0: ()
1: (1)
2: (2)
4: (3)
5: (2,1)
6: (1,2)
8: (4)
9: (3,1)
11: (2,1,1)
12: (1,3)
13: (1,2,1)
16: (5)
17: (4,1)
18: (3,2)
19: (3,1,1)
20: (2,3)
22: (2,1,2)
24: (1,4)
25: (1,3,1)
26: (1,2,2)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], UnsameQ@@First/@Split[stc[#], UnsameQ]&]
CROSSREFS
Positions of distinct (strict) rows in A374515.
Compositions of this type are counted by A374518.
The complement is A374639.
Other types of runs (instead of anti-):
A065120 gives leaders of standard compositions.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
A238424 counts partitions whose first differences are an anti-run.
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 maximal runs:
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 01 2024
STATUS
approved