Formulas and asymptotics of hypergraph Catalan numbers

TL;DR

This paper confirms Gunnells' conjecture on the asymptotic behavior of hypergraph Catalan numbers, revealing that their growth is driven by the enumeration of specific k-tours on star-like trees.

Contribution

The authors provide an alternative proof of Gunnells' conjectured asymptotics for hypergraph Catalan numbers and identify the key role of k-tours on star-like trees in their growth.

Findings

Confirmed Gunnells' asymptotic formula for hypergraph Catalan numbers

Identified the significance of k-tours on star-like trees in asymptotic growth

Provided an alternative enumeration approach

Abstract

Tree walks are a class of closed walks on a complete graph constrained to span trees. In this work, we focus on a special subclass called $k$-tours, which were recently introduced by Gunnells and are enumerated by the hypergraph Catalan numbers $ c_n^{(k)}$. Gunnells conjectured an asymptotic formula for $c_n^{(k)}$ which we confirm through an alternative approach to their enumeration. As it turns out, the asymptotic growth is governed by the number of $k$-tours on star-like trees.