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A362610
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Number of integer partitions of n having a unique part of least multiplicity.
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45
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0, 1, 2, 2, 4, 5, 7, 10, 13, 16, 23, 30, 35, 50, 61, 73, 95, 123, 139, 187, 216, 269, 328, 411, 461, 594, 688, 836, 980, 1211, 1357, 1703, 1936, 2330, 2697, 3253, 3649, 4468, 5057, 6005, 6841, 8182, 9149, 10976, 12341, 14508, 16447, 19380, 21611, 25553, 28628
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OFFSET
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0,3
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COMMENTS
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Alternatively, these are partitions with a part appearing fewer times than each of the others.
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LINKS
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FORMULA
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G.f.: Sum_{m>=2} (Sum_{j>=1} x^(j*(m-1))/(1 + x^(j*m)/(1 - x^j))) * (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 (3,3,2,2,2,1,1,1) has least multiplicity 2, and only one part of multiplicity 2 (namely 3), so is counted under a(15).
The a(1) = 1 through a(8) = 13 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (221) (33) (322) (44)
(211) (311) (222) (331) (332)
(1111) (2111) (411) (511) (422)
(11111) (3111) (2221) (611)
(21111) (4111) (2222)
(111111) (22111) (5111)
(31111) (22211)
(211111) (41111)
(1111111) (221111)
(311111)
(2111111)
(11111111)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Count[Length/@Split[#], Min@@Length/@Split[#]]==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=2, n+1, sum(j=1, n, x^(j*(m-1))/(1 + if(j*m<=n, x^(j*m)/(1-x^j) )) + A)*prod(j=1, n\m, 1 + x^(j*m)/(1 - x^j) + A)), -(n+1)) \\ Andrew Howroyd, May 04 2023
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CROSSREFS
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These partitions have ranks A359178.
A275870 counts collapsible partitions.
A359893 counts partitions by median.
Cf. A008284, A053263, A098859, A237984, A240219, A304442, A327472, A353863, A353864, A353865, A362612.
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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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