Let $\alpha=(a,b,\ldots)$ be a composition. Consider the associated poset $F(\alpha)$, called a fence, whose covering relations are
$$
x_1\lhd x_2 \lhd \ldots\lhd x_{a+1}\rhd x_{a+2}\rhd \ldots\rhd x_{a+b+1}\lhd x_{a+b+2}\lhd \ldots\ .
$$
We study the associated distributive lattice $L(\alpha)$ consisting of all lower order ideals of $F(\alpha)$.
These lattices are important in the theory of cluster algebras and their rank generating functions can be used to define $q$-analogues of rational numbers.
In particular, we make progress on a recent conjecture of Morier-Genoud and Ovsienko that $L(\alpha)$ is rank unimodal. We show that if one of the parts of $\alpha$ is greater than the sum of the others, then the conjecture is true. We conjecture that $L(\alpha)$ enjoys the stronger properties of having a nested chain decomposition and having a rank sequence which is either top or bottom interlacing, the latter being a recently defined property of sequences. We verify that these properties hold for compositions with at most three parts and for what we call $d$-divided posets, generalizing work of Claussen and simplifying a construction of Gansner.