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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.

%I A333628 #6 Mar 31 2020 10:54:22
%S A333628 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,
%T A333628 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,
%U A333628 2,2,3,2,3,3,4,2,2,3,3,2,2,2,4,3,3,4
%N 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.
%C A333628 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.
%C A333628 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.
%H A333628 Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003.
%e A333628 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.
%t A333628 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A333628 runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
%t A333628 Table[runsres[stc[n]],{n,100}]
%Y A333628 Number of times applying A333627 to reach a power of 2, starting with n.
%Y A333628 Positions of first appearances are A333629.
%Y A333628 All of the following pertain to compositions in standard order (A066099):
%Y A333628 - The length is A000120.
%Y A333628 - The partial sums from the right are A048793.
%Y A333628 - The sum is A070939.
%Y A333628 - Adjacent equal pairs are counted by A124762.
%Y A333628 - Equal runs are counted by A124767.
%Y A333628 - Strict compositions are ranked by A233564.
%Y A333628 - The partial sums from the left are A272020.
%Y A333628 - Constant compositions are ranked by A272919.
%Y A333628 - Normal compositions are ranked by A333217.
%Y A333628 - Heinz number is A333219.
%Y A333628 - Anti-runs are counted by A333381.
%Y A333628 - Adjacent unequal pairs are counted by A333382.
%Y A333628 - First appearances for specified run-lengths are A333630.
%Y A333628 Cf. A029931, A098504, A114994, A225620, A228351, A238279, A242882, A318928, A329744, A329747, A333489.
%K A333628 nonn
%O A333628 1,5
%A A333628 _Gus Wiseman_, Mar 31 2020