  For each pair of coprime integers $a$ and $b$ one defines the "rational
$q$-Catalan number" $\mathrm{Cat}(a,b)_q=\bigl[\hskip-1.5pt
\begin{smallmatrix}{a-1+b}\\{a-1}\end{smallmatrix}\hskip-1pt\bigr]_q/[a]_q$. It
is known that this is a polynomial in $q$ with nonnegative integer
coefficients, but this phenomenon is mysterious. Despite recent progress in the
understanding of these polynomials and their two-variable $q,t$-analogues, we
still lack a simple combinatorial interpretation of the coefficients. The
current paper builds on a conjecture of Paul Johnson relating $q$-Catalan
numbers to lattice points. The main idea of this approach is to fix $a$ and
express everything in terms of the weight lattice of type $A_{a-1}$. For a
given $a$ we construct a family of $(a-2)\phi(a)+1$ polynomials called
"$q$-Catalan germs" and for each integer $b$ coprime to $a$ we express
$\mathrm{Cat}(a,b)_q$ in terms of germs. We conjecture that the germs have
nonnegative coefficients and we show that this nonnegativity conjecture is
implied by a stronger conjecture about "ribbon partitions" of certain subposets
of Young's lattice.
