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A325245
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Number of integer partitions of n with adjusted frequency depth 3.
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8
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0, 0, 0, 1, 1, 2, 4, 4, 6, 8, 11, 11, 19, 17, 25, 29, 37, 37, 56, 53, 75, 80, 99, 103, 145, 143, 181, 199, 247, 255, 336, 339, 426, 459, 548, 590, 738, 759, 916, 999, 1192, 1259, 1529, 1609, 1915, 2083, 2406, 2589, 3085, 3267, 3809, 4134, 4763, 5119, 5964
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OFFSET
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0,6
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COMMENTS
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The adjusted frequency depth of an integer partition 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 enumeration of integer partitions by adjusted frequency depth is given by A325280. The adjusted frequency depth of the integer partition with Heinz number n is given by A323014.
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LINKS
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Table of n, a(n) for n=0..54.
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EXAMPLE
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The a(3) = 1 through a(10) = 11 partitions:
(21) (31) (32) (42) (43) (53) (54) (64)
(41) (51) (52) (62) (63) (73)
(321) (61) (71) (72) (82)
(2211) (421) (431) (81) (91)
(521) (432) (532)
(3311) (531) (541)
(621) (631)
(222111) (721)
(3322)
(4321)
(4411)
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MATHEMATICA
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fdadj[ptn_List]:=If[ptn=={}, 0, Length[NestWhileList[Sort[Length/@Split[#]]&, ptn, Length[#]>1&]]];
Table[Length[Select[IntegerPartitions[n], fdadj[#]==3&]], {n, 0, 30}]
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CROSSREFS
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Column k = 3 of A225485 and A325280.
Cf. A008284, A047966, A116608, A127002, A181819, A182850, A323014, A323023, A325239, A325246, A325254, A325268, A325280.
Sequence in context: A339905 A215205 A298043 * A241064 A292671 A210948
Adjacent sequences: A325242 A325243 A325244 * A325246 A325247 A325248
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KEYWORD
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nonn
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AUTHOR
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Gus Wiseman, Apr 15 2019
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STATUS
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approved
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