A351290 Numbers k such that the k-th composition in standard order has all distinct runs.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 50, 51, 52, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 78

Offset: 1

Author

Gus Wiseman, Feb 10 2022

Keywords

• nonn

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.

Examples

			The terms together with their binary expansions and corresponding compositions begin:
   0:      0  ()
   1:      1  (1)
   2:     10  (2)
   3:     11  (1,1)
   4:    100  (3)
   5:    101  (2,1)
   6:    110  (1,2)
   7:    111  (1,1,1)
   8:   1000  (4)
   9:   1001  (3,1)
  10:   1010  (2,2)
  11:   1011  (2,1,1)
  12:   1100  (1,3)
  14:   1110  (1,1,2)
  15:   1111  (1,1,1,1)
		

Links

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

Crossrefs

The version for Heinz numbers and prime multiplicities is A130091.

The version using binary expansions is A175413, complement A351205.

The version for run-lengths instead of runs is A329739.

These compositions are counted by A351013.

The complement is A351291.

A005811 counts runs in binary expansion, distinct A297770.

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:

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

- A351018 = binary expansions, for run-lengths A032020.

- A351200 = patterns, for run-lengths A351292.

- A351202 = permutations of prime factors.

Selected statistics of standard compositions:

- Length is A000120.

- Parts are A066099, reverse A228351.

- Sum is A070939.

- Runs are counted by A124767, distinct A351014.

- 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, A113835, A116608, A238279, A242882, A318928, A325545, A328592, A329745, A350952, A351015, A351201.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],UnsameQ@@Split[stc[#]]&]