A353855 Last term of the trajectory of the composition run-sum transformation (condensation) of the n-th composition in standard order.

0, 1, 2, 2, 4, 5, 6, 4, 8, 9, 8, 8, 12, 13, 8, 8, 16, 17, 18, 18, 20, 17, 22, 20, 24, 25, 24, 24, 20, 17, 18, 16, 32, 33, 34, 34, 32, 37, 38, 32, 40, 41, 32, 34, 44, 45, 32, 40, 48, 49, 50, 50, 52, 49, 54, 52, 40, 41, 40, 32, 32, 37, 34, 32, 64, 65, 66, 66, 68

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, corresponding to (2,1,1) -> (2,2) -> (4), has last term a(11) = 8.

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 139 -> 138 -> 136 -> 128 ends with a(139) = 128.
		

Crossrefs

The version for partitions is A353842.

This trajectory has length A353854, firsts A072639, partitions A353841.

A005811 counts runs in binary expansion.

A011782 counts compositions.

A066099 lists compositions in standard order.

A318928 gives runs-resistance of binary expansion.

A325268 counts partitions by omicron, rank statistic A304465.

A333627 ranks the run-lengths of standard compositions.

A351014 counts distinct runs in standard compositions, firsts A351015.

A353840-A353846 pertain to a partition's run-sum trajectory.

A353847 represents a composition's run-sums, partitions A353832.

A353853-A353859 pertain to a composition's run-sum trajectory.

Cf. A237685, A304442, A333381, A333489, A333755, A353848, A353849, A353850, A353852, A353860.

Programs

Mathematica

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