The following inequality came up while trying to resolve a conjecture about a certain class of partitions (the context is not particularly enlightening): $ B_n^2 \leq B_{n-1}B_{n+1} $ for $n \geq 1$, where $B_n$ denotes the $n$th Bell number (i.e. the number of partitions of an $n$-element set). I ran this inequality through Maple for values of $n$ up to 500 or so and did not find a counterexample.
There is no nice closed form for $B_n$, so I was hoping to prove this inequality combinatorially rather than analytically (particularly since the given inequality is just the simplest version of a more general inequality I hope to establish).
Let $P_n$ be the collection of (not number of) all partitions of an $n$-element set. My approach was to find an injection from $P_n \times P_n$ into $P_{n-1} \times P_{n+1}$. Suppose we are to map $(C_1, D_1)$ to $(C_2, D_2)$ and suppose, for convenience, our ground set is the integers from $1$ to $n$.
A natural seeming choice was to choose $C_2$ to be the partition $C_1$ with the element $n$ removed. Since removing $n$ will map many partitions in $P_n$ to the same partition in $P_{n-1}$, we would need somehow to choose $D_2$ in such a way as to retain information about where $n$ was in $C_1$. We have the new element $n+1$ to work with, so perhaps it can be used to "tag" the partitions in some unique way.
I've stressed a combinatorial approach in this post, but I would greatly appreciate any techniques that might be of use in establishing (or refuting) this inequality.