|
| |
|
|
A353849
|
|
Number of distinct positive run-sums of the n-th composition in standard order.
|
|
34
|
|
|
|
0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 3, 3, 1, 2, 3, 1, 2, 3, 2, 1, 2, 2, 2, 3, 3, 3, 2, 2, 3, 2, 3, 2, 1, 1, 3, 2, 1, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,6
|
|
|
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 sums (4,3,3,4).
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
|
Composition 462903 in standard order is (1,1,4,7,1,2,1,1,1), with run-sums (2,4,7,1,2,3), of which a(462903) = 5 are distinct.
|
|
|
MATHEMATICA
|
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Length[Union[Total/@Split[stc[n]]]], {n, 0, 100}]
|
|
|
CROSSREFS
|
Counting repeated runs also givesA124767.
Positions of first appearances areA246534.
The run-sums themselves are listed byA353932,withA353849distinct terms.
For distinct run-lengths instead of run-sums we haveA354579.
A005811counts runs in binary expansion.
A066099lists compositions in standard order.
A165413counts distinct run-lengths in binary expansion.
A353847represents the run-sum transformation for compositions.
Selected statistics of standard compositions:
- Number of distinct parts isA334028.
Selected classes of standard compositions:
- Constant compositions areA272919.
Cf.A003242,A044813,A071625,A238279,A329738,A333381,A333489,A333755,A353744,A353832,A353850,A353852,A353866.
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
