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A362608
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Number of integer partitions of n having a unique mode.
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57
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0, 1, 2, 2, 4, 5, 7, 11, 16, 21, 29, 43, 54, 78, 102, 131, 175, 233, 295, 389, 490, 623, 794, 1009, 1255, 1579, 1967, 2443, 3016, 3737, 4569, 5627, 6861, 8371, 10171, 12350, 14901, 18025, 21682, 26068, 31225, 37415, 44617, 53258, 63313, 75235, 89173, 105645
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OFFSET
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0,3
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COMMENTS
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A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.
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LINKS
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FORMULA
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G.f.: Sum_{m>=1} (Sum_{j>=1} x^(j*m)*(1 - x^j)/(1 - x^(j*m))) * (Product_{j>=1} (1 - x^(j*m))/(1 - x^j)). - Andrew Howroyd, May 04 2023
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EXAMPLE
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The partition (3,3,2,1) has greatest multiplicity 2, and a unique part of multiplicity 2 (namely 3), so is counted under a(9).
The a(1) = 1 through a(7) = 11 partitions:
(1) (2) (3) (4) (5) (6) (7)
(11) (111) (22) (221) (33) (322)
(211) (311) (222) (331)
(1111) (2111) (411) (511)
(11111) (3111) (2221)
(21111) (3211)
(111111) (4111)
(22111)
(31111)
(211111)
(1111111)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Length[Commonest[#]]==1&]], {n, 0, 30}]
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PROG
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(PARI) seq(n) = my(A=O(x*x^n)); Vec(sum(m=1, n, sum(j=1, n\m, x^(j*m)*(1-x^j)/(1 - x^(j*m)), A)*prod(j=1, n\m, (1 - x^(j*m))/(1 - x^j) + A)/prod(j=n\m+1, n, 1 - x^j + A)), -(n+1)) \\ Andrew Howroyd, May 04 2023
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CROSSREFS
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For parts instead of multiplicities we have A000041(n-1), ranks A102750.
These partitions have ranks A356862.
A275870 counts collapsible partitions.
A359893 counts partitions by median.
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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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