OFFSET
0,2
COMMENTS
Anti-runs summing to n are counted by A003242(n).
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 maximal anti-runs of the 1234567th composition in standard order are ((3,2,1,2),(2,1,2,5,1),(1),(1)), so row 1234567 is (3,2,1,1).
The nonnegative integers, corresponding compositions, and leaders of anti-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)
6: (1,2) -> (1) 21: (2,2,1) -> (2,2)
7: (1,1,1) -> (1,1,1) 22: (2,1,2) -> (2)
8: (4) -> (4) 23: (2,1,1,1) -> (2,1,1)
9: (3,1) -> (3) 24: (1,4) -> (1)
10: (2,2) -> (2,2) 25: (1,3,1) -> (1)
11: (2,1,1) -> (2,1) 26: (1,2,2) -> (1,2)
12: (1,3) -> (1) 27: (1,2,1,1) -> (1,1)
13: (1,2,1) -> (1) 28: (1,1,3) -> (1,1)
14: (1,1,2) -> (1,1) 29: (1,1,2,1) -> (1,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[First/@Split[stc[n], UnsameQ], {n, 0, 100}]
CROSSREFS
Row-leaders of nonempty rows are A065120.
Row-lengths are A333381.
Row-sums are A374516.
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,tabf
AUTHOR
Gus Wiseman, Jul 31 2024
STATUS
approved