

A353849


Number of distinct positive runsums of the nth 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
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OFFSET

0,6


COMMENTS

Every sequence can be uniquely split into a sequence of nonoverlapping 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 kth composition in standard order (graded reverselexicographic, 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 runsums (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 gives A124767.
Positions of first appearances are A246534.
For distinct runs instead of runsums we have A351014 (firsts A351015).
The runsums themselves are listed by A353932, with A353849 distinct terms.
For distinct runlengths instead of runsums we have A354579.
A005811 counts runs in binary expansion.
A066099 lists compositions in standard order.
A165413 counts distinct runlengths in binary expansion.
A353847 represents the runsum transformation for compositions.
Selected statistics of standard compositions:
 Number of distinct parts is A334028.
Selected classes of standard compositions:
 Constant compositions are A272919.
Cf. A003242, A044813, A071625, A238279, A329738, A333381, A333489, A333755, A353744, A353832, A353850, A353852, A353866.


KEYWORD

nonn


AUTHOR



STATUS

approved



