OFFSET
1,1
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.
Also numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed patterns 23-1 or 12-1.
LINKS
EXAMPLE
The sequence together with corresponding compositions begins:
13: (1,2,1)
25: (1,3,1)
27: (1,2,1,1)
29: (1,1,2,1)
41: (2,3,1)
45: (2,1,2,1)
49: (1,4,1)
51: (1,3,1,1)
53: (1,2,2,1)
54: (1,2,1,2)
55: (1,2,1,1,1)
57: (1,1,3,1)
59: (1,1,2,1,1)
61: (1,1,1,2,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], !Greater@@First/@Split[Reverse[stc[#]], LessEqual]&]
- or -
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 300], MatchQ[stc[#], {___, y_, z_, ___, x_, ___}/; x<=y<z]&] (*23-1 or 12-1*)
CROSSREFS
The reverse version is A375295.
A011782 counts compositions.
A374637 counts compositions by sum of leaders of weakly increasing runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Leader is A065120.
- Strict compositions are A233564.
- Constant compositions are A272919.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 13 2024
STATUS
approved