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%I A332340 #6 Feb 18 2020 04:48:05
%S A332340 1,1,1,3,3,4,9,11,13,23,53,78,120,207,357,707,1183,2030,3558,6229,
%T A332340 10868
%N A332340 Number of widely alternately co-strongly normal compositions of n.
%C A332340 An integer partition is widely alternately co-strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) with weakly increasing run-length (co-strong) which, if reversed, are themselves a widely alternately co-strongly normal partition.
%e A332340 The a(1) = 1 through a(8) = 13 compositions:
%e A332340   (1)  (11)  (12)   (121)   (122)    (123)     (1213)     (1232)
%e A332340              (21)   (211)   (212)    (132)     (1231)     (1322)
%e A332340              (111)  (1111)  (1211)   (213)     (1312)     (2123)
%e A332340                             (11111)  (231)     (1321)     (2132)
%e A332340                                      (312)     (2122)     (2312)
%e A332340                                      (321)     (2131)     (2321)
%e A332340                                      (1212)    (2311)     (3122)
%e A332340                                      (2121)    (3121)     (3212)
%e A332340                                      (111111)  (3211)     (12131)
%e A332340                                                (12121)    (13121)
%e A332340                                                (1111111)  (21212)
%e A332340                                                           (122111)
%e A332340                                                           (11111111)
%e A332340 For example, starting with the composition y = (122111) and repeatedly taking run-lengths and reversing gives (122111) -> (321) -> (111). All of these are normal with weakly increasing run-lengths and the last is all 1's, so y is counted under a(8).
%t A332340 totnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[ptn]==Range[Max[ptn]],LessEqual@@Length/@Split[ptn],totnQ[Reverse[Length/@Split[ptn]]]]];
%t A332340 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],totnQ]],{n,0,10}]
%Y A332340 Normal compositions are A107429.
%Y A332340 Compositions with normal run-lengths are A329766.
%Y A332340 The Heinz numbers of the case of partitions are A332290.
%Y A332340 The case of partitions is A332289.
%Y A332340 The total (instead of alternating) version is A332337.
%Y A332340 Not requiring normality gives A332338.
%Y A332340 The strong version is this same sequence.
%Y A332340 The narrow version is a(n) + 1 for n > 1.
%Y A332340 Cf. A181819, A317245, A317491, A329744, A329741, A329746, A332278, A332279, A332292.
%K A332340 nonn,more
%O A332340 0,4
%A A332340 _Gus Wiseman_, Feb 17 2020

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