OFFSET
0,2
COMMENTS
The leaders of maximal 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 58654th composition in standard order is (1,1,3,2,4,1,1,1,2), with maximal weakly increasing runs ((1,1,3),(2,4),(1,1,1,2)), so row 58654 is (1,2,1).
The nonnegative integers, corresponding compositions, and leaders of maximal weakly increasing runs begin:
0: () -> () 15: (1,1,1,1) -> (1)
1: (1) -> (1) 16: (5) -> (5)
2: (2) -> (2) 17: (4,1) -> (4,1)
3: (1,1) -> (1) 18: (3,2) -> (3,2)
4: (3) -> (3) 19: (3,1,1) -> (3,1)
5: (2,1) -> (2,1) 20: (2,3) -> (2)
6: (1,2) -> (1) 21: (2,2,1) -> (2,1)
7: (1,1,1) -> (1) 22: (2,1,2) -> (2,1)
8: (4) -> (4) 23: (2,1,1,1) -> (2,1)
9: (3,1) -> (3,1) 24: (1,4) -> (1)
10: (2,2) -> (2) 25: (1,3,1) -> (1,1)
11: (2,1,1) -> (2,1) 26: (1,2,2) -> (1)
12: (1,3) -> (1) 27: (1,2,1,1) -> (1,1)
13: (1,2,1) -> (1,1) 28: (1,1,3) -> (1)
14: (1,1,2) -> (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], LessEqual], {n, 0, 100}]
CROSSREFS
Row-leaders are A065120.
Row-lengths are A124766.
Row-sums are A374630.
Positions of non-weakly decreasing rows are A375137.
A011782 counts compositions.
A335456 counts patterns matched by compositions.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Leader is A065120.
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Jul 20 2024
STATUS
approved