%I #5 Feb 15 2020 11:05:02

%S 1,1,1,2,2,1,2,2,1,1,3,1,1,1,2,2,1,1,1,1,2,2,1,1,1,1,1,1,3,1,1,1,1,1,

%T 1,2,2,1,1,1,1,1,2,1,1,2,1,1,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1,1,1,1,2,1,

%U 1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,2,1,1

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

%C An integer partition is widely alternately co-strongly normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly increasing run-lengths (co-strong) which, if reversed, are themselves a widely alternately co-strongly normal partition.

%e The a(1) = 1, a(3) = 2, and a(10) = 3 partitions:

%e (1) (21) (4321)

%e (111) (322111)

%e (1111111111)

%e For example, starting with y = (4,3,2,2,1,1,1) and repeatedly taking run-lengths and reversing gives y -> (3,2,1,1) -> (2,1,1) -> (2,1) -> (1,1). These are all normal, have weakly increasing run-lengths, and the last is all 1's, so y is counted a(14).

%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[IntegerPartitions[n],totnQ]],{n,0,30}]

%Y Normal partitions are A000009.

%Y Dominated by A317245.

%Y The non-co-strong version is A332277.

%Y The total (instead of alternate) version is A332278.

%Y The Heinz numbers of these partitions are A332290.

%Y The strong version is A332292.

%Y The case of reversed partitions is (also) A332292.

%Y The generalization to compositions is A332340.

%Y Cf. A100883, A107429, A133808, A316496, A317081, A317256, A317491, A329746, A332291, A332295, A332297, A332337, A332338, A332339.

%K nonn

%O 0,4

%A _Gus Wiseman_, Feb 13 2020