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%I #9 Jun 26 2020 06:05:46
%S 1,1,2,3,2,3,3,2,2,2,3,3,2,2,3,3,2,2,2,2,2,3,2,2,2,2,2,2,3,2,3,2,2,2,
%T 2,2,3,2,3,2,2,2,2,2,2,3,2,2,2,2,2,2,2,2,2,4,3,2,2,2,2,2,2,2,2,2,3,2,
%U 2,2,2,2,2,2,2,2,2,2,3,2,2
%N Number of narrowly totally strongly normal integer partitions of n.
%C A partition is narrowly totally strongly normal if either it is empty, a singleton (narrow), or it covers an initial interval of positive integers (normal) and has weakly decreasing run-lengths (strong) that are themselves a narrowly totally strongly normal partition.
%e The a(1) = 1, a(2) = 2, a(3) = 3, and a(55) = 4 partitions:
%e (1) (2) (3) (55)
%e (1,1) (2,1) (10,9,8,7,6,5,4,3,2,1)
%e (1,1,1) (5,5,5,5,5,4,4,4,4,3,3,3,2,2,1)
%e (1)^55
%e For example, starting with the partition (3,3,2,2,1) and repeatedly taking run-lengths gives (3,3,2,2,1) -> (2,2,1) -> (2,1) -> (1,1) -> (2). The first four are normal and have weakly decreasing run-lengths, and the last is a singleton, so (3,3,2,2,1) is counted under a(11).
%t tinQ[q_]:=Or[q=={},Length[q]==1,And[Union[q]==Range[Max[q]],GreaterEqual@@Length/@Split[q],tinQ[Length/@Split[q]]]];
%t Table[Length[Select[IntegerPartitions[n],tinQ]],{n,0,30}]
%Y Normal partitions are A000009.
%Y The non-totally normal version is A316496.
%Y The widely alternating version is A332292.
%Y The non-strong case of compositions is A332296.
%Y The case of compositions is A332336.
%Y The wide version is a(n) - 1 for n > 1.
%Y Cf. A001462, A025487, A100883, A181819, A182850, A317081, A317245, A317256, A317491, A332275, A332277, A332278, A332291, A332337.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Feb 15 2020
%E a(60)-a(80) from _Jinyuan Wang_, Jun 26 2020
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