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A332337 Number of widely totally strongly normal compositions of n.

%I A332337 #5 Feb 16 2020 07:55:18
%S A332337 1,1,1,3,3,3,9,9,12,23,54,77,116,205,352,697,1174,2013,3538,6209,10830
%N A332337 Number of widely totally strongly normal compositions of n.
%C A332337 A sequence is widely totally strongly normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly decreasing run-lengths (strong) that are themselves a widely totally strongly normal sequence.
%F A332337 For n > 1, a(n) = A332336(n) - 1.
%e A332337 The a(1) = 1 through a(8) = 12 compositions:
%e A332337   (1)  (11)  (12)   (112)   (212)    (123)     (1213)     (1232)
%e A332337              (21)   (121)   (221)    (132)     (1231)     (2123)
%e A332337              (111)  (1111)  (11111)  (213)     (1312)     (2132)
%e A332337                                      (231)     (1321)     (2312)
%e A332337                                      (312)     (2131)     (2321)
%e A332337                                      (321)     (3121)     (3212)
%e A332337                                      (1212)    (11221)    (12131)
%e A332337                                      (2121)    (12121)    (13121)
%e A332337                                      (111111)  (1111111)  (21212)
%e A332337                                                           (22112)
%e A332337                                                           (111221)
%e A332337                                                           (11111111)
%e A332337 For example, starting with (22112) and repeated taking run-lengths gives (22112) -> (221) -> (21) -> (11). These are all normal with weakly decreasing run-lengths, and the last is all 1's, so (22112) is counted under a(8).
%t A332337 totnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[ptn]==Range[Max[ptn]],LessEqual@@Length/@Split[ptn],totnQ[Length/@Split[ptn]]]];
%t A332337 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],totnQ]],{n,0,10}]
%Y A332337 Normal compositions are A107429.
%Y A332337 The case of partitions is A332278.
%Y A332337 The non-strong version is A332279.
%Y A332337 Heinz numbers in the case of partitions are A332291.
%Y A332337 The narrow version is A332336.
%Y A332337 The alternating version is A332340.
%Y A332337 The co-strong version is this same sequence.
%Y A332337 Cf. A025487, A100883, A181819, A317245, A317491, A329744, A332274, A332276, A332277, A332292, A332293, A332296.
%K A332337 nonn,more
%O A332337 0,4
%A A332337 _Gus Wiseman_, Feb 15 2020