A354579 Number of distinct lengths of runs in the n-th composition in standard order.

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

Offset: 0

Author

Gus Wiseman, Jun 11 2022

Keywords

• nonn

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 lengths (2,3,1,2).

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.

Examples

			The positions of first appearances together with the corresponding compositions begin:
       1: (1)
      11: (2,1,1)
     119: (1,1,2,1,1,1)
    5615: (2,2,1,1,1,2,1,1,1,1)
  251871: (1,1,1,2,2,1,1,1,1,2,1,1,1,1,1)
		

Crossrefs

Standard compositions are listed by A066099.

The version for partitions is A071625.

For runs instead of run-lengths we have A351014, firsts A351015.

Positions of 0's and 1's are A353744, counted by A329738.

For sums instead of lengths we have A353849, ones at A353848.

Positions of first appearances are A354906.

A003242 counts anti-run compositions, ranked by A333489.

A005811 counts runs in binary expansion.

A333627 ranks the run-lengths of standard compositions.

A351596 ranks compositions with distinct run-lengths, counted by A329739.

A353835 counts distinct run-sums of prime indices, weak A353861.

A353847 ranks the run-sums of standard compositions.

A353852 ranks compositions with distinct run-sums, counted by A353850.

A353860 counts collapsible compositions.

Cf. A029837, A124767, A181819, A238279, A333381, A333755, A353839, A353851.

Programs

Mathematica

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