What's the largest number of maximal chains contained within any poset with $n$ elements?
My best guess is http://oeis.org/A000792 (take a partition $$\coprod A_i = P$$ and let $a < b$ if $a \in A_i, b \in A_j, i < j$, then the number of maximal chains is the product of partition sizes) but I cannot prove it.
This question was mistakenly posed in a module test without solution.
From other places I got the answer as yes.
Let $f(x) =$ max length of chain ending in $x$, $h$ = max length of chain in $P$. We will construct an injection from the set of maximal chains $\{C_1, C_2, ...\}$ into $ \prod_{i=0}^{h} f^{-1}(i) $ by inserting incomparable elements into the chain for each i. Since $f$ is monotone, it is injective when restricted to each $C_k$.
For each $i$ such that $i \notin f(C_k)$, every element in $f^{-1}(i)$ is incomparable with $C_k$ by maximality of $C_k$. There is an element $x$ comparable with $\{y \in C_k: f(y) > i\}$ so x is incomparable with $\{y \in C_k: f(y) < i\}$. The injection is given by inserting such elements into $C_k$. We choose this x because then the inverse function is clear, given a sequence of elements ordered by increasing f, we remove an element if it is incomparable with all previous unremoved elements.
Thus the number of maximal chains <= max product of sizes of a partition, and this is a tight bound from the question