Catalan number sequences and generalized action graphs

TL;DR

This paper explores the relationship between various Catalan number sequences and action graphs, showing limitations of current generalizations and proposing a conjecture for super Catalan numbers.

Contribution

It demonstrates that action graphs cannot be extended to several Catalan-related sequences and proposes a new approach for super Catalan numbers.

Findings

Action graphs relate to Catalan numbers and Fuss-Catalan numbers.

Action graphs cannot be generalized to Catalan's triangle, (a,b)-Catalan numbers, or internal triangles.

A conjecture is proposed for constructing action graphs for super Catalan numbers.

Abstract

Action graphs emerged from work of Bergner and Hackney on category actions in the context of Reedy categories. Alvarez, Bergner, and Lopez showed that action graphs could be inductively generated without reference to category actions and have a close relationship with the sequence of Catalan numbers. These graphs were further generalized in work of Cressman, Lin, Nguyen, and Wiljanen, who showed that the Fuss-Catalan numbers have a similar relation to another set of inductively defined directed graphs. In our paper, we consider several other sequences related to the Catalan numbers, namely Catalan's triangle, $(a,b)$-Catalan numbers, internal triangles, and super Catalan numbers. We show action graphs cannot be generalized to Catalan's triangle, $(a,b)$-Catalan numbers, nor internal triangles. We also conjecture a method for constructing action graphs for the Super Catalan numbers.