A290518 Minimum value of Product_{i in lambda} i!, where lambda ranges over all partitions of n into distinct parts.

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

Author

Alois P. Heinz, Aug 04 2017

Keywords

• nonn

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..960

Crossrefs

Cf. A000142, A000217, A004736, A290517.

Programs

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 *)

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 }.