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A332296 Number of narrowly totally normal compositions of n.

%I A332296 #7 Feb 16 2020 07:54:59
%S A332296 1,1,2,4,5,7,13,23,30,63,120,209,369,651,1198,2174,3896,7023,12699,
%T A332296 22941,41565
%N A332296 Number of narrowly totally normal compositions of n.
%C A332296 A sequence is narrowly totally normal if either it is empty, a singleton (narrow), or it covers an initial interval of positive integers (normal) with narrowly totally normal run-lengths.
%C A332296 A composition of n is a finite sequence of positive integers summing to n.
%F A332296 For n > 1, a(n) = A332279(n) + 1.
%e A332296 The a(0) = 1 through a(6) = 13 compositions:
%e A332296   ()  (1)  (2)   (3)    (4)     (5)      (6)
%e A332296            (11)  (12)   (112)   (122)    (123)
%e A332296                  (21)   (121)   (212)    (132)
%e A332296                  (111)  (211)   (221)    (213)
%e A332296                         (1111)  (1121)   (231)
%e A332296                                 (1211)   (312)
%e A332296                                 (11111)  (321)
%e A332296                                          (1212)
%e A332296                                          (1221)
%e A332296                                          (2112)
%e A332296                                          (2121)
%e A332296                                          (11211)
%e A332296                                          (111111)
%e A332296 For example, starting with the composition (1,1,2,3,1,1) and repeatedly taking run-lengths gives (1,1,2,3,1,1) -> (2,1,1,2) -> (1,2,1) -> (1,1,1) -> (3). The first four are normal and the last is a singleton, so (1,1,2,3,1,1) is counted under a(9).
%t A332296 tinQ[q_]:=Or[Length[q]<=1,And[Union[q]==Range[Max[q]],tinQ[Length/@Split[q]]]];
%t A332296 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],tinQ]],{n,0,10}]
%Y A332296 Normal compositions are A107429.
%Y A332296 The wide version is A332279.
%Y A332296 The wide recursive version (for partitions) is A332295.
%Y A332296 The alternating version is A332296 (this sequence).
%Y A332296 The strong version is A332336.
%Y A332296 The co-strong version is (also) A332336.
%Y A332296 Cf. A001462, A316496, A317081, A317245, A317491, A329744, A332276, A332277, A332278, A332297, A332337, A332340.
%K A332296 nonn,more
%O A332296 0,3
%A A332296 _Gus Wiseman_, Feb 15 2020