OFFSET
0,4
COMMENTS
A sequence is widely totally strongly normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly decreasing run-lengths (strong) that are themselves a widely totally strongly normal sequence.
FORMULA
For n > 1, a(n) = A332336(n) - 1.
EXAMPLE
The a(1) = 1 through a(8) = 12 compositions:
(1) (11) (12) (112) (212) (123) (1213) (1232)
(21) (121) (221) (132) (1231) (2123)
(111) (1111) (11111) (213) (1312) (2132)
(231) (1321) (2312)
(312) (2131) (2321)
(321) (3121) (3212)
(1212) (11221) (12131)
(2121) (12121) (13121)
(111111) (1111111) (21212)
(22112)
(111221)
(11111111)
For example, starting with (22112) and repeated taking run-lengths gives (22112) -> (221) -> (21) -> (11). These are all normal with weakly decreasing run-lengths, and the last is all 1's, so (22112) is counted under a(8).
MATHEMATICA
totnQ[ptn_]:=Or[ptn=={}, Union[ptn]=={1}, And[Union[ptn]==Range[Max[ptn]], LessEqual@@Length/@Split[ptn], totnQ[Length/@Split[ptn]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], totnQ]], {n, 0, 10}]
CROSSREFS
Normal compositions are A107429.
The case of partitions is A332278.
The non-strong version is A332279.
Heinz numbers in the case of partitions are A332291.
The narrow version is A332336.
The alternating version is A332340.
The co-strong version is this same sequence.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Feb 15 2020
STATUS
approved