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A332277
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Number of widely totally normal integer partitions of n.
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14
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1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 4, 4, 2, 4, 4, 6, 3, 5, 7, 6, 8, 12, 9, 12, 13, 11, 12, 18, 17, 12, 32, 19, 25, 33, 30, 28, 44, 33, 43, 57, 51, 60, 83, 70, 83, 103, 96, 97, 125, 117, 134, 157, 157, 171, 226, 215, 238, 278, 302, 312, 359, 357, 396, 450, 444, 477, 580
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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.
Also the number of widely totally normal reversed integer partitions of n.
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LINKS
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EXAMPLE
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The a(n) partitions for n = 1, 4, 10, 11, 16, 18:
1 211 4321 33221 443221 543321
1111 33211 322211 4432111 4333221
322111 332111 1111111111111111 4432221
1111111111 11111111111 4433211
43322211
44322111
111111111111111111
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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[IntegerPartitions[n], recnQ]], {n, 0, 30}]
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CROSSREFS
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Taking multiplicities instead of run-lengths gives A317245.
Constantly recursively normal partitions are A332272.
The Heinz numbers of these partitions are A332276.
The case of all compositions (not just partitions) is A332279.
The narrow version is a(n) + 1 for n > 1.
Cf. A181819, A316496, A317081, A317256, A317491, A317588, A329746, A329747, A332289, A332290, A332291, A332296, A332297, A332336, A332337, A332340.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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