A351596 Numbers k such that the k-th composition in standard order has all distinct run-lengths.

0, 1, 2, 3, 4, 7, 8, 10, 11, 14, 15, 16, 19, 21, 23, 26, 28, 30, 31, 32, 35, 36, 39, 42, 47, 56, 60, 62, 63, 64, 67, 71, 73, 74, 79, 84, 85, 87, 95, 100, 106, 112, 119, 120, 122, 123, 124, 126, 127, 128, 131, 135, 136, 138, 143, 146, 159, 164, 168, 170, 171

Offset: 1

Author

Gus Wiseman, Feb 24 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)
   7:    111  (1,1,1)
   8:   1000  (4)
  10:   1010  (2,2)
  11:   1011  (2,1,1)
  14:   1110  (1,1,2)
  15:   1111  (1,1,1,1)
  16:  10000  (5)
  19:  10011  (3,1,1)
  21:  10101  (2,2,1)
  23:  10111  (2,1,1,1)
		

Crossrefs

The version using binary expansions is A044813.

The version for Heinz numbers and prime multiplicities is A130091.

These compositions are counted by A329739, normal A329740.

The version for runs instead of run-lengths is A351290, counted by A351013.

A005811 counts runs in binary expansion, distinct A297770.

A011782 counts integer compositions.

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 run-lengths:

- A032020 = binary expansions, for runs A351018.

- A351017 = binary words, for runs A351016.

- A351292 = patterns, for runs A351200.

Selected statistics of standard compositions (A066099, A228351):

- Length is A000120.

- Sum is A070939.

- Runs are counted by A124767, distinct A351014.

- Heinz number is A333219.

- Number of distinct parts is A334028.

Cf. A098859, A106356, A175413, A238279, A242882, A328592, A329745, A333628, A350952, A351015, A351202.

Programs

Mathematica

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