A353849 Number of distinct positive run-sums of the n-th composition in standard order.

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

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

Table of n, a(n) for n=0..86.

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

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 gives A124767.

Positions of first appearances are A246534.

For distinct runs instead of run-sums we have A351014 (firsts A351015).

A version for partitions is A353835, weak A353861.

Positions of 1's are A353848, counted by A353851.

The version for binary expansion is A353929 (firsts A353930).

The run-sums themselves are listed by A353932, with A353849 distinct terms.

For distinct run-lengths instead of run-sums we have A354579.

A005811 counts runs in binary expansion.

A066099 lists compositions in standard order.

A165413 counts distinct run-lengths in binary expansion.

A297770 counts distinct runs in binary expansion, firsts A350952.

A353847 represents the run-sum transformation for compositions.

A353853-A353859 pertain to composition run-sum trajectory.

Distinct runs: A032020, A175413, A351013, A351018, A329739, A351290.

Selected statistics of standard compositions:

- Length is A000120.

- Sum is A070939.

- Heinz number is A333219.

- Number of distinct parts is A334028.

Selected classes of standard compositions:

- Partitions are A114994, strict A333256.

- Multisets are A225620, strict A333255.

- Strict compositions are A233564.

- Constant compositions are A272919.

Cf. A003242, A044813, A071625, A238279, A329738, A333381, A333489, A333755, A353744, A353832, A353850, A353852, A353866.

Sequence in context: A269972 A202205 A336123 * A075661 A206829 A319694

Adjacent sequences: A353846 A353847 A353848 * A353850 A353851 A353852

Keyword

nonn

Author

Gus Wiseman, May 30 2022

Status

approved