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A325831
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Number of integer partitions of n whose number of submultisets is greater than n.
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10
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1, 1, 1, 2, 2, 4, 5, 8, 10, 16, 21, 35, 40, 58, 84, 120, 141, 199, 255, 347, 447, 592, 772, 1006, 1172, 1504, 1928, 2455, 3061, 3859, 4778, 5953, 7054, 8737, 10742, 13193, 15783, 19241, 23412, 28344, 33951, 40911, 49150, 58917, 70482, 84055, 100069, 118914
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OFFSET
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0,4
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COMMENTS
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The number of submultisets of a partition is the product of its multiplicities, each plus one.
The Heinz numbers of these partitions are given by A325795.
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LINKS
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FORMULA
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EXAMPLE
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The a(1) = 1 through a(8) = 10 partitions:
(1) (11) (21) (211) (221) (321) (421) (3221)
(111) (1111) (311) (2211) (2221) (3311)
(2111) (3111) (3211) (4211)
(11111) (21111) (4111) (22211)
(111111) (22111) (32111)
(31111) (41111)
(211111) (221111)
(1111111) (311111)
(2111111)
(11111111)
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MAPLE
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b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
`if`(n=p-1, 1, 0), add(`if`(irem(p, j+1, 'r')=0,
(w-> b(w, min(w, i-1), r))(n-i*j), 0), j=0..n/i))
end:
a:= n-> combinat[numbpart](n)-add(b(n$2, k), k=0..n):
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Times@@(1+Length/@Split[#])>n&]], {n, 0, 30}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1,
If[n == p - 1, 1, 0], Sum[If[Mod[p, j + 1] == 0,
Function[w, b[w, Min[w, i-1], p/(j+1)]][n-i*j], 0], {j, 0, n/i}]];
a[n_] := PartitionsP[n] - Sum[b[n, n, k], {k, 0, n}];
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CROSSREFS
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Cf. A002033, A098859, A126796, A325694, A325792, A325795, A325828, A325830, A325832, A325833, A325834, A325836.
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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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