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

0,3

Links

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

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

Cf. A000142, A000217, A004736, A290517.

Sequence in context: A054481 A262501 A225422 * A242882 A361426 A157285

Adjacent sequences: A290515 A290516 A290517 * A290519 A290520 A290521

Keyword

nonn

Author

Alois P. Heinz, Aug 04 2017

Status

approved