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A332336 Number of narrowly totally strongly normal compositions of n.

%I A332336 #6 Feb 16 2020 07:55:12
%S A332336 1,1,2,4,4,4,10,10,13,24,55,78,117,206,353,698,1175,2014,3539,6210,
%T A332336 10831
%N A332336 Number of narrowly totally strongly normal compositions of n.
%C A332336 A sequence 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 sequence.
%C A332336 A composition of n is a finite sequence of positive integers summing to n.
%F A332336 For n > 1, a(n) = A332337(n) + 1.
%e A332336 The a(1) = 1 through a(8) = 13 compositions:
%e A332336   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A332336        (11)  (12)   (112)   (212)    (123)     (1213)     (1232)
%e A332336              (21)   (121)   (221)    (132)     (1231)     (2123)
%e A332336              (111)  (1111)  (11111)  (213)     (1312)     (2132)
%e A332336                                      (231)     (1321)     (2312)
%e A332336                                      (312)     (2131)     (2321)
%e A332336                                      (321)     (3121)     (3212)
%e A332336                                      (1212)    (11221)    (12131)
%e A332336                                      (2121)    (12121)    (13121)
%e A332336                                      (111111)  (1111111)  (21212)
%e A332336                                                           (22112)
%e A332336                                                           (111221)
%e A332336                                                           (11111111)
%e A332336 For example, starting with (22112) and repeated taking run-lengths gives (22112) -> (221) -> (21) -> (11) -> (2). The first four are normal with weakly decreasing run-lengths, and the last is a singleton, so (22112) is counted under a(8).
%t A332336 tinQ[q_]:=Or[q=={},Length[q]==1,And[Union[q]==Range[Max[q]],GreaterEqual@@Length/@Split[q],tinQ[Length/@Split[q]]]];
%t A332336 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],tinQ]],{n,0,10}]
%Y A332336 Normal compositions are A107429.
%Y A332336 The non-strong version is A332296.
%Y A332336 The case of partitions is A332297.
%Y A332336 The co-strong version is A332336 (this sequence).
%Y A332336 The wide version is A332337.
%Y A332336 Cf. A025487, A316496, A317081, A317245, A317256, A317491, A329744, A332279, A332291, A332292, A332338, A332340.
%K A332336 nonn,more
%O A332336 0,3
%A A332336 _Gus Wiseman_, Feb 15 2020