Monotony of partition functions
Consider the partitions of a natural number with respect to a
given subset A of the natural numbers.
Bateman and Erdos (1956) asked the question, for which sets A
the number of these partitions is monotonically increasing. They gave a
simple criterion for asymptotic monotonicity.
In 1994 Friedman, Joichi, and Stanton classified these sets assuming
a basic monotonicity conjecture. A recent proof of this conjecture
(Prellberg and Stanton, 2003) provides a complete answer to the question.