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A332279
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Number of widely totally normal compositions of n.
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8
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1, 1, 1, 3, 4, 6, 12, 22, 29, 62, 119, 208, 368, 650, 1197, 2173, 3895, 7022, 12698, 22940, 41564
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OFFSET
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0,4
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COMMENTS
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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.
A composition of n is a finite sequence of positive integers with sum n.
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LINKS
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FORMULA
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EXAMPLE
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The a(1) = 1 through a(7) = 22 compositions:
(1) (11) (12) (112) (122) (123) (1123)
(21) (121) (212) (132) (1132)
(111) (211) (221) (213) (1213)
(1111) (1121) (231) (1231)
(1211) (312) (1312)
(11111) (321) (1321)
(1212) (2113)
(1221) (2122)
(2112) (2131)
(2121) (2212)
(11211) (2311)
(111111) (3112)
(3121)
(3211)
(11221)
(12112)
(12121)
(12211)
(21121)
(111211)
(112111)
(1111111)
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).
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MATHEMATICA
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recnQ[ptn_]:=Or[ptn=={}, Union[ptn]=={1}, And[Union[ptn]==Range[Max[ptn]], recnQ[Length/@Split[ptn]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], recnQ]], {n, 0, 10}]
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CROSSREFS
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Constantly recursively normal partitions are A332272.
The case of reversed partitions is (also) A332277.
The co-strong version is (also) A332337.
Cf. A001462, A181819, A182850, A317081, A317245, A317491, A329744, A332276, A332289, A332292, A332295, A332297, A332336, A332340.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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