Generalized Path Pairs and Fuss-Catalan Triangles

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.