OFFSET
0,12
COMMENTS
Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with lengths (2,3,1,2).
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.
EXAMPLE
The positions of first appearances together with the corresponding compositions begin:
1: (1)
11: (2,1,1)
119: (1,1,2,1,1,1)
5615: (2,2,1,1,1,2,1,1,1,1)
251871: (1,1,1,2,2,1,1,1,1,2,1,1,1,1,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Length[Union[Length/@Split[stc[n]]]], {n, 0, 100}]
CROSSREFS
Standard compositions are listed by A066099.
The version for partitions is A071625.
Positions of first appearances are A354906.
A005811 counts runs in binary expansion.
A333627 ranks the run-lengths of standard compositions.
A353847 ranks the run-sums of standard compositions.
A353860 counts collapsible compositions.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 11 2022
STATUS
approved