OFFSET
0,3
COMMENTS
A sequence is narrowly totally normal if either it is empty, a singleton (narrow), or it covers an initial interval of positive integers (normal) with narrowly totally normal run-lengths.
A composition of n is a finite sequence of positive integers summing to n.
FORMULA
For n > 1, a(n) = A332279(n) + 1.
EXAMPLE
The a(0) = 1 through a(6) = 13 compositions:
() (1) (2) (3) (4) (5) (6)
(11) (12) (112) (122) (123)
(21) (121) (212) (132)
(111) (211) (221) (213)
(1111) (1121) (231)
(1211) (312)
(11111) (321)
(1212)
(1221)
(2112)
(2121)
(11211)
(111111)
For example, starting with the composition (1,1,2,3,1,1) and repeatedly taking run-lengths gives (1,1,2,3,1,1) -> (2,1,1,2) -> (1,2,1) -> (1,1,1) -> (3). The first four are normal and the last is a singleton, so (1,1,2,3,1,1) is counted under a(9).
MATHEMATICA
tinQ[q_]:=Or[Length[q]<=1, And[Union[q]==Range[Max[q]], tinQ[Length/@Split[q]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], tinQ]], {n, 0, 10}]
CROSSREFS
Normal compositions are A107429.
The wide version is A332279.
The wide recursive version (for partitions) is A332295.
The alternating version is A332296 (this sequence).
The strong version is A332336.
The co-strong version is (also) A332336.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Feb 15 2020
STATUS
approved