A353854 Length of the trajectory of the composition run-sum transformation (condensation) of the n-th composition in standard order.

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

Offset: 0

Author

Gus Wiseman, Jun 01 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 sums (4,3,3,4). The run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353847) until the rank of an anti-run is reached. For example, the trajectory 11 -> 10 -> 8 corresponds to the trajectory (2,1,1) -> (2,2) -> (4), with length a(11) = 3.

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 trajectory of 94685 and the a(94685) = 5 corresponding compositions:
  94685: (2,1,1,4,1,1,2,1,1,2,1)
  86357: (2,2,4,2,2,2,2,1)
  69889: (4,4,8,1)
  65793: (8,8,1)
  65537: (16,1)
		

Crossrefs

Positions of first appearances are A072639.

Positions of 1's are A333489, counted by A003242 (complement A261983).

The version for partitions is A353841.

The last part of the same trajectory is A353855.

This is the rank statistic counted by A353859.

A005811 counts runs in binary expansion.

A011782 counts compositions.

A066099 lists compositions in standard order.

A318928 gives runs-resistance of binary expansion.

A333627 represents the run-lengths of standard compositions.

A353832 represents the run-sum transformation of a partition.

A353840-A353846 pertain to the partition run-sum trajectory.

A353847 represents the run-sum transformation of a composition.

A353853-A353859 pertain to the composition run-sum trajectory.

A353932 lists run-sums of standard compositions, represented by A353847.

Cf. A237685, A238279, A304442, A325277, A333755, A353848, A353849, A353850, A353852, A353860.

Programs

Mathematica

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