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A266871
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Number of partitions of n that maximize the product of multiplicities of parts.
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3
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1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1
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OFFSET
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0,9
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LINKS
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FORMULA
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EXAMPLE
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a(8) = 2: [1,1,1,1,1,1,1,1], [1,1,1,1,2,2] (product of multiplicities = 8).
a(9) = 1: [1,1,1,1,1,2,2] (product = 10).
a(10) = 2: [1,1,1,1,1,1,2,2], [1,1,1,1,2,2,2] (product = 12).
a(11) = 1: [1,1,1,1,1,2,2,2] (product = 15).
a(23) = 3: [1,1,1,1,1,1,1,1,1,2,2,2,2,3,3], [1,1,1,1,1,1,1,1,2,2,2,3,3,3], [1,1,1,1,1,1,2,2,2,2,3,3,3] (product = 72).
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MAPLE
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b:= proc(n, i) option remember; local r, l, j;
if n=0 or i=1 then [max(1, n), 1]
else r:= b(n, i-1);
for j to iquo(n, i) do
l:= (w-> [w[1]*j, w[2]])(b(n-i*j, i-1));
r:= `if`(l[1]>r[1], l,
`if`(l[1]=r[1], [0, l[2]], 0)+r)
od; r
fi
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..120);
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MATHEMATICA
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b[n_, i_] := b[n, i] = Module[{r, l, j}, If[n == 0 || i == 1, {Max[1, n], 1}, r = b[n, i - 1]; For[j = 1, j <= Quotient[n, i], j++, l = Function[w, {w[[1]]*j, w[[2]]}][b[n - i*j, i - 1]]; r = If[l[[1]] > r[[1]], l, If[l[[1]] == r[[1]], {0, l[[2]]}, 0] + r]]; r]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 120}] (* Jean-François Alcover, Dec 21 2016, translated from Maple *)
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CROSSREFS
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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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