OFFSET
1,1
COMMENTS
First differs from the non-dashed version in lacking 166, corresponding to the composition (2,3,1,2).
The leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.
Also numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed patterns 1-32 or 1-21.
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)
45: (2,1,2,1)
49: (1,4,1)
50: (1,3,2)
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)
77: (3,1,2,1)
82: (2,3,2)
89: (2,1,3,1)
91: (2,1,2,1,1)
93: (2,1,1,2,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], !Greater@@First/@Split[stc[#], LessEqual]&]
- or -
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], MatchQ[stc[#], {___, x_, ___, z_, y_, ___}/; x<=y<z]&] (*1-32 or 1-21*)
CROSSREFS
For leaders of identical runs we have A335485.
The reverse version is A375296.
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 12 2024
STATUS
approved