A333628 Runs-resistance of the n-th composition in standard order. Number of steps taking run-lengths to reduce the n-th composition in standard order to a singleton.

0, 0, 1, 0, 2, 2, 1, 0, 2, 1, 3, 2, 2, 3, 1, 0, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 4, 3, 4, 3, 1, 0, 2, 2, 3, 1, 2, 2, 3, 2, 2, 1, 2, 2, 2, 3, 3, 2, 2, 2, 4, 2, 3, 2, 4, 3, 4, 2, 3, 3, 4, 3, 1, 0, 2, 2, 3, 2, 2, 2, 3, 2, 3, 3, 4, 2, 2, 3, 3, 2, 2, 2, 4, 3, 3, 4

Offset: 1

Author

Gus Wiseman, Mar 31 2020

Keywords

• nonn

Comments

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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.

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.

Examples

			Starting with 13789 and repeatedly applying A333627 gives: 13789 -> 859 -> 110 -> 29 -> 11 -> 6 -> 3 -> 2, corresponding to the compositions: (1,2,2,1,1,2,1,1,2,1) -> (1,2,2,1,2,1,1) -> (1,2,1,1,2) -> (1,1,2,1) -> (2,1,1) -> (1,2) -> (1,1) -> (2), so a(13789) = 7.
		

Links

• Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Crossrefs

Number of times applying A333627 to reach a power of 2, starting with n.

Positions of first appearances are A333629.

All of the following pertain to compositions in standard order (A066099):

- The length is A000120.

- The partial sums from the right are A048793.

- The sum is A070939.

- Adjacent equal pairs are counted by A124762.

- Equal runs are counted by A124767.

- Strict compositions are ranked by A233564.

- The partial sums from the left are A272020.

- Constant compositions are ranked by A272919.

- Normal compositions are ranked by A333217.

- Heinz number is A333219.

- Anti-runs are counted by A333381.

- Adjacent unequal pairs are counted by A333382.

- First appearances for specified run-lengths are A333630.

Cf. A029931, A098504, A114994, A225620, A228351, A238279, A242882, A318928, A329744, A329747, A333489.

Programs

Mathematica

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