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A329743
Number of compositions of n with runs-resistance n - 3.
2
0, 0, 0, 1, 2, 6, 9, 16, 8
OFFSET
0,5
COMMENTS
A composition of n is a finite sequence of positive integers with sum n.
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.
EXAMPLE
The a(3) = 1 through a(8) = 8 compositions:
(3) (22) (14) (114) (1123) (12113)
(1111) (23) (411) (1132) (12212)
(32) (1113) (1141) (13112)
(41) (1221) (1411) (21131)
(131) (2112) (2122) (21221)
(212) (3111) (2212) (31121)
(11112) (2311) (121112)
(11211) (3211) (211121)
(21111) (11131)
(11212)
(11221)
(12211)
(13111)
(21211)
(111121)
(121111)
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).
MATHEMATICA
runsres[q_]:=If[Length[q]==1, 0, Length[NestWhileList[Length/@Split[#]&, q, Length[#]>1&]]-1];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], runsres[#]==n-3&]], {n, 10}]
CROSSREFS
Column k = n - 3 of A329744.
Column k = 3 of A329750.
Compositions with runs-resistance 2 are A329745.
Sequence in context: A345051 A265202 A355738 * A320496 A172433 A049622
KEYWORD
nonn,fini,full
AUTHOR
Gus Wiseman, Nov 21 2019
STATUS
approved