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1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1
a(71)-a(77) from Jinyuan Wang, Jun 26 2020
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For example, starting with the partition y = (4,4,4,3,3,2,1) and repeatedly taking run-lengths and reversing gives (4,4,4,3,3,2,1) -> (1,1,2,3) -> (1,1,2) -> (1,2) -> (1,1). All of these are normal with weakly increasing decreasing run-lengths, and the last is all 1's, so y is counted under a(21).
For example, starting with the partition y = (4,4,4,3,3,2,1) and repeatedly taking run-lengths and reversing gives (4,4,4,3,3,2,1) -> (1,1,2,3) -> (1,1,2) -> (1,2) -> (1,1). All of these are normal with weakly increasing run-lengths, and the last is all 1's, so y is counted under a(21).
allocated for Gus WisemanNumber of widely alternately strongly normal integer partitions of n.
1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2
0,4
An integer partition is widely alternately strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly decreasing run-lengths (strong) which, if reversed, are themselves a widely alternately strongly normal partition.
Also the number of widely alternately co-strongly normal reversed integer partitions of n.
The a(1) = 1, a(3) = 2, and a(21) = 3 partitions:
(1) (21) (654321)
(111) (4443321)
(111111111111111111111)
totnQ[ptn_]:=Or[ptn=={}, Union[ptn]=={1}, And[Union[ptn]==Range[Max[ptn]], GreaterEqual@@Length/@Split[ptn], totnQ[Reverse[Length/@Split[ptn]]]]];
Table[Length[Select[IntegerPartitions[n], totnQ]], {n, 0, 30}]
Normal partitions are A000009.
The non-strong version is A332277.
The co-strong version is A332289.
The case of reversed partitions is (also) A332289.
The Heinz numbers of these partitions are A332291.
The case of compositions is A332340.
Cf. A100883, A181819, A317081, A317245, A317256, A317491, A329746, A329747, A332278, A332290, A332297, A332337.
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Gus Wiseman, Feb 16 2020
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