A351014 Number of distinct runs in the n-th composition in standard order.

0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 1, 1, 2, 2, 2, 1, 3, 3, 2, 2, 3, 1, 2, 3, 2, 2, 2, 2, 2, 3, 3, 3, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 1, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 2, 2, 3

Offset

0,6

Comments

The n-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 n, 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

The number 3310 has binary expansion 110011101110 and standard composition (1,3,1,1,2,1,1,2), with runs (1), (3), (1,1), (2), (1,1), (2), of which 4 are distinct, so a(3310) = 4.

Mathematica

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Length[Union[Split[stc[n]]]], {n, 0, 100}]

Crossrefs

Counting not necessarily distinct runs gives A124767.

Using binary expansions instead of standard compositions gives A297770.

Positions of first appearances are A351015.

A005811 counts runs in binary expansion.

A011782 counts integer compositions.

A044813 lists numbers whose binary expansion has distinct run-lengths.

A085207 represents concatenation of standard compositions, reverse A085208.

A333489 ranks anti-runs, complement A348612.

A345167 ranks alternating compositions, counted by A025047.

A351204 counts partitions where every permutation has all distinct runs.

Counting words with all distinct runs:

- A351013 = compositions, for run-lengths A329739, ranked by A351290.

- A351016 = binary words, for run-lengths A351017.

- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.

- A351200 = patterns, for run-lengths A351292.

- A351202 = permutations of prime factors.

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. A098859, A106356, A116608, A238279, A242882, A318928, A325545, A328592, A329745, A350952, A351201.

Sequence in context: A236479 A116514 A334028 * A124767 A319443 A130633

Adjacent sequences: A351011 A351012 A351013 * A351015 A351016 A351017

Keyword

nonn

Author

Gus Wiseman, Feb 07 2022

Status

approved