  Path pairs are a modification of parallelogram polyominoes that provide yet
another combinatorial interpretation of the Catalan numbers. More generally,
the number of path pairs of length $n$ and distance $\delta$ corresponds to the
$(n-1,\delta-1)$ entry of Shapiro's so-called Catalan triangle. In this paper,
we widen the notion of path pairs $(\gamma_1,\gamma_2)$ to the situation where
$\gamma_1$ and $\gamma_2$ may have different lengths, and then enforce
divisibility conditions on runs of vertical steps in $\gamma_2$. This creates a
two-parameter family of integer triangles that generalize the Catalan triangle
and qualify as proper Riordan arrays for many choices of parameters. In
particular, we use generalized path pairs to provide a new combinatorial
interpretation for all entries in every proper Riordan array
$\mathcal{R}(d(t),h(t))$ of the form $d(t) = C_k(t)^i$, $h(t) = t \kern+1pt
C_k(t)^k$, where $1 \leq i \leq k$ and $C_k(t)$ is the generating function for
some sequence of Fuss-Catalan numbers (some $k \geq 2$). Closed formulas are
then provided for the number of generalized path pairs across an even broader
range of parameters, as well as for the number of weak path pairs with a fixed
number of non-initial intersections.

Combinatorics