OFFSET
0,4
COMMENTS
A sequence is widely totally normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has widely totally normal run-lengths.
A composition of n is a finite sequence of positive integers with sum n.
FORMULA
For n > 1, a(n) = A332296(n) - 1.
EXAMPLE
The a(1) = 1 through a(7) = 22 compositions:
(1) (11) (12) (112) (122) (123) (1123)
(21) (121) (212) (132) (1132)
(111) (211) (221) (213) (1213)
(1111) (1121) (231) (1231)
(1211) (312) (1312)
(11111) (321) (1321)
(1212) (2113)
(1221) (2122)
(2112) (2131)
(2121) (2212)
(11211) (2311)
(111111) (3112)
(3121)
(3211)
(11221)
(12112)
(12121)
(12211)
(21121)
(111211)
(112111)
(1111111)
For example, starting with y = (3,2,1,1,2,2,2,1,2,1,1,1,1) and repeatedly taking run-lengths gives y -> (1,1,2,3,1,1,4) -> (2,1,1,2,1) -> (1,2,1,1) -> (1,1,2) -> (2,1) -> (1,1). These are all normal and the last is all 1's, so y is counted under a(20).
MATHEMATICA
recnQ[ptn_]:=Or[ptn=={}, Union[ptn]=={1}, And[Union[ptn]==Range[Max[ptn]], recnQ[Length/@Split[ptn]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], recnQ]], {n, 0, 10}]
CROSSREFS
Normal compositions are A107429.
Constantly recursively normal partitions are A332272.
The case of partitions is A332277.
The case of reversed partitions is (also) A332277.
The narrow version is A332296.
The strong version is A332337.
The co-strong version is (also) A332337.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Feb 12 2020
STATUS
approved