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A290518
Minimum value of Product_{i in lambda} i!, where lambda ranges over all partitions of n into distinct parts.
2
1, 1, 2, 2, 6, 12, 12, 48, 144, 288, 288, 1440, 5760, 17280, 34560, 34560, 207360, 1036800, 4147200, 12441600, 24883200, 24883200, 174182400, 1045094400, 5225472000, 20901888000, 62705664000, 125411328000, 125411328000, 1003290624000, 7023034368000
OFFSET
0,3
LINKS
FORMULA
a(n) = A000142(n) / A290517(n).
a(0) = 1, a(n) = a(n-1) * A004736(n) for n>0.
a(n) = a(n-1) iff n in { A000217 } \ { 0 }.
EXAMPLE
a(10) = 288 = (4! * 3! * 2! * 1!) is the value for partition [4,3,2,1]. All other partitions of 10 into distinct parts give larger values: [5,3,2]-> 1440, [5,4,1]-> 2880, [6,3,1]-> 4320, [6,4]-> 17280, [7,2,1]-> 10080, [7,3]-> 30240, [8,2]-> 80640, [9,1]-> 362880, [10]-> 3628800.
MAPLE
b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, infinity,
`if`(n=0, 1, min(b(n, i-1), b(n-i, min(n-i, i-1))*i!)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, a(n-1)*
(t-> t*(t+3)/2-n+2)(floor(sqrt(8*n-7)/2-1/2)))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Aug 05 2017
MATHEMATICA
b[n_, i_]:=b[n, i]=If[n>i*(i + 1)/2, Infinity, If[n==0, 1, Min[b[n, i - 1], b[n - i, Min[n - i, i - 1]]*i!]]]; Table[b[n, n], {n, 0, 30}] (* Indranil Ghosh, Aug 05 2017, after Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 04 2017
STATUS
approved