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Formulas and asymptotics of hypergraph Catalan numbers

Eva-Maria Hainzl

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.

Formulas and asymptotics of hypergraph Catalan numbers

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 -tours, which were recently introduced by Gunnells and are enumerated by the hypergraph Catalan numbers . Gunnells conjectured an asymptotic formula for which we confirm through an alternative approach to their enumeration. As it turns out, the asymptotic growth is governed by the number of -tours on star-like trees.

Paper Structure

This paper contains 3 sections, 7 theorems, 68 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Counting $k$-tours
  3. Asympotics of the hypergraph Catalan numbers

Key Result

Theorem 1

Let $k\geq 1$ and $c_n^{(k)}$ be the number of $k$-tours on trees with $n+1$ vertices. Then and for $k\geq 3$, as $n\rightarrow \infty$.

Figures (4)

  • Figure 1: The tree walk $W = (1,2,1,3,4,3,5,3,4,3,1,2,1,3,6,3,5,6,3,1)$ separated into outgoing steps at vertices and into sequence of vertices
  • Figure 2: Decomposition of a 2-tour in $\mathcal{F}$. The root is colored red and its only child $c$ green. Steps in set $A$ are purple and steps in set $B$ are dark blue.
  • Figure 3: Stars: the only rooted trees contributing to the asymptotic growth of $c_n^{(k)}$ for $k>2$. (The root is marked green)
  • Figure 4: Star-like trees: rooted trees contributing to the asymptotic growth of $c_n^{(2)}$. (The root is marked green)

Theorems & Definitions (16)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • proof : Proof of Theorem \ref{['lem:closed_form']}
  • Lemma 1
  • proof
  • proof : Alternative proof of Theorem \ref{['lem:closed_form']}
  • Theorem 1
  • Definition 1: Stars and star-like trees
  • Proposition 1
  • ...and 6 more