OFFSET
0,4
COMMENTS
An integer partition is widely alternately co-strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) with weakly increasing run-length (co-strong) which, if reversed, are themselves a widely alternately co-strongly normal partition.
EXAMPLE
The a(1) = 1 through a(8) = 13 compositions:
(1) (11) (12) (121) (122) (123) (1213) (1232)
(21) (211) (212) (132) (1231) (1322)
(111) (1111) (1211) (213) (1312) (2123)
(11111) (231) (1321) (2132)
(312) (2122) (2312)
(321) (2131) (2321)
(1212) (2311) (3122)
(2121) (3121) (3212)
(111111) (3211) (12131)
(12121) (13121)
(1111111) (21212)
(122111)
(11111111)
For example, starting with the composition y = (122111) and repeatedly taking run-lengths and reversing gives (122111) -> (321) -> (111). All of these are normal with weakly increasing run-lengths and the last is all 1's, so y is counted under a(8).
MATHEMATICA
totnQ[ptn_]:=Or[ptn=={}, Union[ptn]=={1}, And[Union[ptn]==Range[Max[ptn]], LessEqual@@Length/@Split[ptn], totnQ[Reverse[Length/@Split[ptn]]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], totnQ]], {n, 0, 10}]
CROSSREFS
Normal compositions are A107429.
Compositions with normal run-lengths are A329766.
The Heinz numbers of the case of partitions are A332290.
The case of partitions is A332289.
The total (instead of alternating) version is A332337.
Not requiring normality gives A332338.
The strong version is this same sequence.
The narrow version is a(n) + 1 for n > 1.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Feb 17 2020
STATUS
approved