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Insertion algorithms and pattern avoidance on trees arising in the Kapranov embedding of $\overline{M}_{0,n+3}$

Andrew Reimer-Berg

TL;DR

This work provides a complete combinatorial resolution to the question of bijecting two tree models, $\mathrm{Tour}(\underline{k})$ and $\mathrm{Slide}^{\omega}(\underline{k})$, that encode intersection numbers on $\overline{M}_{0,n+3}$ via psi and omega classes under forgetting maps. The author constructs an explicit insertion-based bijection using maps $\hat{\sigma}_{i,j}$ and $\hat{\sigma}_{j}$ and a unifying insertion framework $\Sigma_{\underline{k}}$, with a clearly defined inverse. A parallel development characterizes caterpillar slide trees by pattern-avoidance criteria, showing that caterpillar words correspond to $2{-}1{-}2$ and $23{-}\overline{2}{-}1$-avoiding words under suitable constraints, and introducing the BigRep/TotalRep conditions to distinguish $\psi$- versus $\omega$-slides. In the all-ones case, the paper gives a direct bijection between $\mathrm{Slide}^{\omega}(1,\dots,1)$ and $S_n$ via two explicit maps, yielding a complete, direct combinatorial narrative that mirrors the geometric equalities of the associated intersection numbers. Overall, the results illuminate the combinatorial underpinnings of the Kapranov embedding and its pullbacks of $\psi$ classes, framing a precise bridge between tree combinatorics and asymmetric multinomial coefficients.

Abstract

We resolve a question of Gillespie, Griffin, and Levinson that asks for a combinatorial bijection between two classes of trivalent trees, tournament trees and slide trees, that both naturally arise in the intersection theory of the moduli space $\overline{M}_{0,n+3}$ of stable genus zero curves with $n+3$ marked points. Each set of trees enumerates the same intersection product of certain pullbacks of $ψ$ classes under forgetting maps. We give an explicit combinatorial bijection between these two sets of trees using an insertion algorithm. We also classify the words that appear on the slide trees of caterpillar shape via pattern avoidance conditions.

Insertion algorithms and pattern avoidance on trees arising in the Kapranov embedding of $\overline{M}_{0,n+3}$

TL;DR

This work provides a complete combinatorial resolution to the question of bijecting two tree models, and , that encode intersection numbers on via psi and omega classes under forgetting maps. The author constructs an explicit insertion-based bijection using maps and and a unifying insertion framework , with a clearly defined inverse. A parallel development characterizes caterpillar slide trees by pattern-avoidance criteria, showing that caterpillar words correspond to and -avoiding words under suitable constraints, and introducing the BigRep/TotalRep conditions to distinguish - versus -slides. In the all-ones case, the paper gives a direct bijection between and via two explicit maps, yielding a complete, direct combinatorial narrative that mirrors the geometric equalities of the associated intersection numbers. Overall, the results illuminate the combinatorial underpinnings of the Kapranov embedding and its pullbacks of classes, framing a precise bridge between tree combinatorics and asymmetric multinomial coefficients.

Abstract

We resolve a question of Gillespie, Griffin, and Levinson that asks for a combinatorial bijection between two classes of trivalent trees, tournament trees and slide trees, that both naturally arise in the intersection theory of the moduli space of stable genus zero curves with marked points. Each set of trees enumerates the same intersection product of certain pullbacks of classes under forgetting maps. We give an explicit combinatorial bijection between these two sets of trees using an insertion algorithm. We also classify the words that appear on the slide trees of caterpillar shape via pattern avoidance conditions.

Paper Structure

This paper contains 21 sections, 50 theorems, 34 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Background
  4. The moduli space of curves $\overline{M}_{0,n}$
  5. Psi and omega classes, the Kapranov embedding, and multidegrees
  6. Asymmetric multinomial coefficients
  7. Slide trees
  8. Pattern Avoidance
  9. Preliminaries on combinatorics of slide trees
  10. The main bijection
  11. Preliminaries
  12. The map $\hat{\sigma}_{i,j}$
  13. The map $\hat{\sigma}_{j}$
  14. Constructing the full bijection
  15. An explicit example of the bijection
  16. ...and 6 more sections

Key Result

Proposition 1.1

Let $\underline{k}$ be a composition of $n$. Then, and

Figures (2)

  • Figure 1: An element of $\overline{M}_{0,n+3}$ for $n=4$, along with its dual tree.
  • Figure 2: A slide tree in $\mathrm{Slide}^\omega(1,0,1,1,1,1,1,1,2,1,1,1)$ with the labels $\geq3$ incremented by 1. In this case, $j=7$.

Theorems & Definitions (162)

  • Proposition 1.1: From GGL22 and GGL23
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Example 2.4
  • Definition 2.5
  • Example 2.6
  • ...and 152 more