Generalized Path Pairs and Fuss-Catalan Triangles

TL;DR

This paper introduces a broad generalization of path pairs related to Catalan numbers, establishing new combinatorial interpretations for Riordan arrays involving Fuss-Catalan numbers and deriving explicit formulas for their counts.

Contribution

It extends the concept of path pairs to include different lengths and divisibility conditions, linking them to a wide class of Riordan arrays and Fuss-Catalan number sequences.

Findings

Generalized path pairs correspond to entries in new integer triangles.

Provides combinatorial interpretations for all entries in certain Riordan arrays.

Derives closed formulas for counting generalized and weak path pairs.

Abstract

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.