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A329976
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Number of partitions p of n such that (number of numbers in p that have multiplicity 1) > (number of numbers in p having multiplicity > 1).
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8
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0, 1, 1, 2, 2, 3, 4, 6, 9, 14, 18, 27, 38, 50, 66, 89, 113, 145, 186, 234, 297, 374, 468, 585, 737, 912, 1140, 1407, 1758, 2153, 2668, 3254, 4007, 4855, 5946, 7170, 8705, 10451, 12626, 15068, 18125, 21551, 25766, 30546, 36365, 42958, 50976, 60062, 70987
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OFFSET
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0,4
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COMMENTS
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For each partition of n, let
d = number of terms that are not repeated;
r = number of terms that are repeated.
a(n) is the number of partitions such that d > r.
Also the number of integer partitions of n with median multiplicity 1. - Gus Wiseman, Mar 20 2023
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LINKS
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FORMULA
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EXAMPLE
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The partitions of 6 are 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111.
These have d > r: 6, 51, 42, 321
These have d = r: 411, 3222, 21111
These have d < r: 33, 222, 2211, 111111
Thus, a(6) = 4.
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MATHEMATICA
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z = 30; d[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] == 1 &]]];
r[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; Table[Count[IntegerPartitions[n], p_ /; d[p] > r[p]], {n, 0, z}]
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CROSSREFS
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For parts instead of multiplicities we have A027336
The complement is counted by A330001.
A116608 counts partitions by number of distinct parts.
A237363 counts partitions with median difference 0.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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