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

%I A290518 #14 Aug 05 2017 18:34:10
%S A290518 1,1,2,2,6,12,12,48,144,288,288,1440,5760,17280,34560,34560,207360,
%T A290518 1036800,4147200,12441600,24883200,24883200,174182400,1045094400,
%U A290518 5225472000,20901888000,62705664000,125411328000,125411328000,1003290624000,7023034368000
%N A290518 Minimum value of Product_{i in lambda} i!, where lambda ranges over all partitions of n into distinct parts.
%H A290518 Alois P. Heinz, <a href="/A290518/b290518.txt">Table of n, a(n) for n = 0..960</a>
%F A290518 a(n) = A000142(n) / A290517(n).
%F A290518 a(0) = 1, a(n) = a(n-1) * A004736(n) for n>0.
%F A290518 a(n) = a(n-1) iff n in { A000217 } \ { 0 }.
%e A290518 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.
%p A290518 b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, infinity,
%p A290518      `if`(n=0, 1, min(b(n, i-1), b(n-i, min(n-i, i-1))*i!)))
%p A290518     end:
%p A290518 a:= n-> b(n$2):
%p A290518 seq(a(n), n=0..30);
%p A290518 # second Maple program:
%p A290518 a:= proc(n) option remember; `if`(n=0, 1, a(n-1)*
%p A290518       (t-> t*(t+3)/2-n+2)(floor(sqrt(8*n-7)/2-1/2)))
%p A290518     end:
%p A290518 seq(a(n), n=0..30);  # _Alois P. Heinz_, Aug 05 2017
%t A290518 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 *)
%Y A290518 Cf. A000142, A000217, A004736, A290517.
%K A290518 nonn
%O A290518 0,3
%A A290518 _Alois P. Heinz_, Aug 04 2017