A353853 Trajectory of the composition run-sum transformation (or condensation) of n, using standard composition numbers.

0, 1, 2, 3, 2, 4, 5, 6, 7, 4, 8, 9, 10, 8, 11, 10, 8, 12, 13, 14, 10, 8, 15, 8, 16, 17, 18, 19, 18, 20, 21, 17, 22, 23, 20, 24, 25, 26, 24, 27, 26, 24, 28, 20, 29, 21, 17, 30, 18, 31, 16, 32, 33, 34, 35, 34, 36, 32, 37, 38, 39, 36, 32, 40, 41, 42, 32

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 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.

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 given in row 11 corresponds to the trajectory (2,1,1) -> (2,2) -> (4).

Examples

			Triangle begins:
   0
   1
   2
   3  2
   4
   5
   6
   7  4
   8
   9
  10  8
  11 10  8
  12
  13
  14 10  8
For example, the trajectory of 29 is 29 -> 21 -> 17, corresponding to the compositions (1,1,2,1) -> (2,2,1) -> (4,1).
		

Crossrefs

These sequences for partitions are A353840-A353846.

This is the iteration of A353847, with partition version A353832.

Row-lengths are A353854, counted by A353859.

Final terms are A353855.

Counting rows by weight of final term gives A353856.

Rows ending in a power of 2 are A353857, counted by A353858.

A003242 counts anti-run compositions, ranked by A333489, complement A261983.

A005811 counts runs in binary expansion.

A011782 counts compositions.

A066099 lists compositions in standard order.

A318928 gives runs-resistance of binary expansion.

A329739 counts compositions with all distinct run-lengths.

A333627 ranks the run-lengths of standard compositions.

A351014 counts distinct runs in standard compositions, firsts A351015.

A353853-A353859 pertain to composition run-sum trajectory.

A353929 counts distinct runs in binary expansion, firsts A353930.

A353932 lists run-sums of standard compositions.

Cf. A237685, A238279, A304442, A325277, A333381, A333755, A353833, A353848, A353849, A353850, A353852, A353860.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[NestWhileList[stcinv[Total/@Split[stc[#]]]&,n,MatchQ[stc[#],{_,x_,x_,_}]&],{n,0,50}]