[seqfan] Re: Is a partition with distinct parts and maximum product of parts, unique ?

[Brendan McKay]

Note that if the number of parts is specified as well as the sum, then
either there is no such partition, or the unique partition into distinct
parts maximizing the product is the partition with no gap or one
gap of size 1.

Such as (4 parts):  2+3+4+5=14, 2+3+4+6=15, 2+3+5+6=16,
  2+4+5+6=17, 3+4+5+6=18, 3+4+5+7=19.

The proof is trivial: if there are two gaps or one gap of size 2 or
more, make a larger product using  (a+1)(b-1) > ab for b-a>2.

Brendan.