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%I #8 Mar 09 2020 18:26:18
%S 1,1,1,3,4,6,12,22,29,62,119,208,368,650,1197,2173,3895,7022,12698,
%T 22940,41564
%N Number of widely totally normal compositions of n.
%C A sequence is widely totally normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has widely totally normal run-lengths.
%C A composition of n is a finite sequence of positive integers with sum n.
%F For n > 1, a(n) = A332296(n) - 1.
%e The a(1) = 1 through a(7) = 22 compositions:
%e (1) (11) (12) (112) (122) (123) (1123)
%e (21) (121) (212) (132) (1132)
%e (111) (211) (221) (213) (1213)
%e (1111) (1121) (231) (1231)
%e (1211) (312) (1312)
%e (11111) (321) (1321)
%e (1212) (2113)
%e (1221) (2122)
%e (2112) (2131)
%e (2121) (2212)
%e (11211) (2311)
%e (111111) (3112)
%e (3121)
%e (3211)
%e (11221)
%e (12112)
%e (12121)
%e (12211)
%e (21121)
%e (111211)
%e (112111)
%e (1111111)
%e For example, starting with y = (3,2,1,1,2,2,2,1,2,1,1,1,1) and repeatedly taking run-lengths gives y -> (1,1,2,3,1,1,4) -> (2,1,1,2,1) -> (1,2,1,1) -> (1,1,2) -> (2,1) -> (1,1). These are all normal and the last is all 1's, so y is counted under a(20).
%t recnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[ptn]==Range[Max[ptn]],recnQ[Length/@Split[ptn]]]];
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],recnQ]],{n,0,10}]
%Y Normal compositions are A107429.
%Y Constantly recursively normal partitions are A332272.
%Y The case of partitions is A332277.
%Y The case of reversed partitions is (also) A332277.
%Y The narrow version is A332296.
%Y The strong version is A332337.
%Y The co-strong version is (also) A332337.
%Y Cf. A001462, A181819, A182850, A317081, A317245, A317491, A329744, A332276, A332289, A332292, A332295, A332297, A332336, A332340.
%K nonn,more
%O 0,4
%A _Gus Wiseman_, Feb 12 2020
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