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A325836
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Number of integer partitions of n having n - 1 different submultisets.
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11
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0, 0, 0, 1, 1, 2, 0, 3, 0, 5, 2, 2, 0, 15, 0, 2, 3, 25, 0, 17, 0, 18, 3, 2, 0, 150, 0, 2, 13, 24, 0, 43, 0, 351, 3, 2, 2, 383, 0, 2, 3, 341, 0, 60, 0, 37, 51, 2, 0, 3733, 0, 31, 3, 42, 0, 460, 1, 633, 3, 2, 0, 1780, 0, 2, 68, 12460, 0, 87, 0, 55, 3
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OFFSET
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0,6
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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 A325694.
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LINKS
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EXAMPLE
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The a(3) = 1 through a(13) = 15 partitions (empty columns not shown):
(3) (22) (32) (322) (432) (3322) (32222) (4432)
(41) (331) (531) (4411) (71111) (5332)
(511) (621) (5422)
(3222) (5521)
(6111) (6322)
(6331)
(6511)
(7411)
(8221)
(8311)
(9211)
(33322)
(55111)
(322222)
(811111)
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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-> b(n$2, n-1):
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Times@@(1+Length/@Split[#])==n-1&]], {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, r = p/(j + 1);
Function[w, b[w, Min[w, i - 1], r]][n - i*j], 0], {j, 0, n/i}]];
a[n_] := b[n, n, n-1];
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CROSSREFS
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Cf. A002033, A088880, A088881, A108917, A325694, A325768, A325792, A325798, A325828, A325830, A325833, A325835.
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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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