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 A332289 Number of widely alternately co-strongly normal integer partitions of n. 11
 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, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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. LINKS EXAMPLE The a(1) = 1, a(3) = 2, and a(10) = 3 partitions:   (1)  (21)   (4321)        (111)  (322111)               (1111111111) 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). 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[IntegerPartitions[n], totnQ]], {n, 0, 30}] CROSSREFS Normal partitions are A000009. Dominated by A317245. The non-co-strong version is A332277. The total (instead of alternate) version is A332278. The Heinz numbers of these partitions are A332290. The strong version is A332292. The case of reversed partitions is (also) A332292. The generalization to compositions is A332340. Cf. A100883, A107429, A133808, A316496, A317081, A317256, A317491, A329746, A332291, A332295, A332297, A332337, A332338, A332339. Sequence in context: A014710 A055174 A096369 * A102297 A269570 A243759 Adjacent sequences:  A332286 A332287 A332288 * A332290 A332291 A332292 KEYWORD nonn AUTHOR Gus Wiseman, Feb 13 2020 STATUS approved

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Last modified January 21 10:25 EST 2022. Contains 350477 sequences. (Running on oeis4.)