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%I #6 Nov 21 2019 22:14:59
%S 0,0,0,1,2,6,9,16,8
%N Number of compositions of n with runs-resistance n - 3.
%C A composition of n is a finite sequence of positive integers with sum n.
%C 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.
%e The a(3) = 1 through a(8) = 8 compositions:
%e (3) (22) (14) (114) (1123) (12113)
%e (1111) (23) (411) (1132) (12212)
%e (32) (1113) (1141) (13112)
%e (41) (1221) (1411) (21131)
%e (131) (2112) (2122) (21221)
%e (212) (3111) (2212) (31121)
%e (11112) (2311) (121112)
%e (11211) (3211) (211121)
%e (21111) (11131)
%e (11212)
%e (11221)
%e (12211)
%e (13111)
%e (21211)
%e (111121)
%e (121111)
%e For example, repeatedly taking run-lengths starting with (1,2,1,1,3) gives (1,2,1,1,3) -> (1,1,2,1) -> (2,1,1) -> (1,2) -> (1,1) -> (2), which is 5 steps, and 5 = 8 - 3, so (1,2,1,1,3) is counted under a(8).
%t runsres[q_]:=If[Length[q]==1,0,Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1];
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==n-3&]],{n,10}]
%Y Column k = n - 3 of A329744.
%Y Column k = 3 of A329750.
%Y Compositions with runs-resistance 2 are A329745.
%Y Cf. A000740, A008965, A098504, A242882, A318928, A329746, A329747, A329767.
%K nonn,fini,full
%O 0,5
%A _Gus Wiseman_, Nov 21 2019
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