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Is it true for sufficiently large integers $x, y$ that $p(x+y) < p(x)p(y)$, where $p(n)$ is the number of integer partitions of $n$?
Proving an Inequality Involving Integer Partitions
In this post, they have presented an elegant solution via Ferrers graph for the inequality $p(a+b+ab)>p(a)p(b)$. I tried applying this technique to the question above unsuccessfully. Am I overlooking a simple argument?
This inequality appeared in a somewhat recent article of Ono and Bessenrodt. See Theorem 2.1 of "Maximal multiplicative properties of partitions" in Annals of Combinatorics Volume 20 (2016). The article is also available on arxiv: https://arxiv.org/abs/1403.3352.
Their proof is through asymptotics for $p(n)$, so it doesn't give you a "simple" proof. But yes, it is a known inequality.
