OFFSET
1,1
COMMENTS
The leaders of maximal 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.
EXAMPLE
The sequence of terms together with the corresponding compositions begins:
3: (1,1)
7: (1,1,1)
10: (2,2)
14: (1,1,2)
15: (1,1,1,1)
21: (2,2,1)
23: (2,1,1,1)
27: (1,2,1,1)
28: (1,1,3)
29: (1,1,2,1)
30: (1,1,1,2)
31: (1,1,1,1,1)
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 non-distinct (or non-strict) rows in A374515.
Compositions of this type are counted by A374678.
A065120 gives leaders of standard compositions.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
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 06 2024
STATUS
approved