A329743 - OEIS

%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