%I #6 Feb 18 2020 04:48:05

%S 1,1,1,3,3,4,9,11,13,23,53,78,120,207,357,707,1183,2030,3558,6229,

%T 10868

%N Number of widely alternately co-strongly normal compositions of n.

%C 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.

%e The a(1) = 1 through a(8) = 13 compositions:

%e (1) (11) (12) (121) (122) (123) (1213) (1232)

%e (21) (211) (212) (132) (1231) (1322)

%e (111) (1111) (1211) (213) (1312) (2123)

%e (11111) (231) (1321) (2132)

%e (312) (2122) (2312)

%e (321) (2131) (2321)

%e (1212) (2311) (3122)

%e (2121) (3121) (3212)

%e (111111) (3211) (12131)

%e (12121) (13121)

%e (1111111) (21212)

%e (122111)

%e (11111111)

%e 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).

%t totnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[ptn]==Range[Max[ptn]],LessEqual@@Length/@Split[ptn],totnQ[Reverse[Length/@Split[ptn]]]]];

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],totnQ]],{n,0,10}]

%Y Normal compositions are A107429.

%Y Compositions with normal run-lengths are A329766.

%Y The Heinz numbers of the case of partitions are A332290.

%Y The case of partitions is A332289.

%Y The total (instead of alternating) version is A332337.

%Y Not requiring normality gives A332338.

%Y The strong version is this same sequence.

%Y The narrow version is a(n) + 1 for n > 1.

%Y Cf. A181819, A317245, A317491, A329744, A329741, A329746, A332278, A332279, A332292.

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Feb 17 2020