  Let alpha = (a,b,...) be a composition. Consider the associated poset
F(alpha), called a fence, whose covering relations are x_1 < x_2 < ... <
x_{a+1} > x_{a+2} > ... > x_{a+b+1} < x_{a+b+2} < ... . 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.
