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A325334
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Number of integer partitions of n with adjusted frequency depth 3 whose parts cover an initial interval of positive integers.
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10
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0, 0, 0, 1, 0, 0, 2, 0, 0, 1, 1, 0, 2, 0, 0, 2, 0, 0, 2, 0, 1, 2, 0, 0, 2, 0, 0, 1, 1, 0, 4, 0, 0, 1, 0, 0, 3, 0, 0, 1, 1, 0, 3, 0, 0, 3, 0, 0, 2, 0, 1, 1, 0, 0, 2, 1, 1, 1, 0, 0, 4, 0, 0, 2, 0, 0, 3, 0, 0, 1, 1, 0, 3, 0, 0, 2, 0, 0, 3, 0, 1, 1, 0, 0, 4, 0, 0
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OFFSET
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0,7
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COMMENTS
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The adjusted frequency depth of an integer partition (A325280) is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).
The Heinz numbers of these partitions are given by A325374.
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LINKS
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FORMULA
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EXAMPLE
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The first 30 terms count the following partitions:
3: (21)
6: (321)
6: (2211)
9: (222111)
10: (4321)
12: (332211)
12: (22221111)
15: (54321)
15: (2222211111)
18: (333222111)
18: (222222111111)
20: (44332211)
21: (654321)
21: (22222221111111)
24: (333322221111)
24: (2222222211111111)
27: (222222222111111111)
28: (7654321)
30: (5544332211)
30: (444333222111)
30: (333332222211111)
30: (22222222221111111111)
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MATHEMATICA
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normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];
unifQ[m_]:=SameQ@@Length/@Split[m];
Table[Length[Select[IntegerPartitions[n], normQ[#]&&!SameQ@@#&&unifQ[#]&]], {n, 0, 30}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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