OFFSET
1,1
COMMENTS
The leaders of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly 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 terms together with corresponding compositions begin:
26: (1,2,2)
50: (1,3,2)
53: (1,2,2,1)
58: (1,1,2,2)
90: (2,1,2,2)
98: (1,4,2)
100: (1,3,3)
101: (1,3,2,1)
106: (1,2,2,2)
107: (1,2,2,1,1)
114: (1,1,3,2)
117: (1,1,2,2,1)
122: (1,1,1,2,2)
154: (3,1,2,2)
164: (2,3,3)
178: (2,1,3,2)
181: (2,1,2,2,1)
186: (2,1,1,2,2)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], !GreaterEqual@@First/@Split[stc[#], Less]&]
CROSSREFS
For leaders of identical runs we have A335485.
Ranked by positions of non-weakly decreasing rows in A374683.
The complement is counted by A374697.
Compositions of this type are counted by A375135.
A374700 counts compositions by sum of leaders of strictly increasing runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Leader is A065120.
- Parts are listed by A066099.
- Strict compositions are A233564.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 12 2024
STATUS
approved