Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An elaborate new proof of Cayley's formula

Esther Banaian, Anh Trong Nam Hoang, Elizabeth Kelley, Weston Miller, Jason Stack, Carolyn Stephen, Nathan Williams

Abstract

We construct a bijection between certain Deodhar components of a braid variety constructed from an affine Kac-Moody group of type $A_{n-1}$ and vertex-labeled trees on $n$ vertices. By an argument of Galashin, Lam, and Williams using Opdam's trace formula in the affine Hecke algebra and an identity due to Haglund, we obtain an elaborate new proof for the enumeration of the number of vertex-labeled trees on $n$ vertices.

An elaborate new proof of Cayley's formula

Abstract

We construct a bijection between certain Deodhar components of a braid variety constructed from an affine Kac-Moody group of type and vertex-labeled trees on vertices. By an argument of Galashin, Lam, and Williams using Opdam's trace formula in the affine Hecke algebra and an identity due to Haglund, we obtain an elaborate new proof for the enumeration of the number of vertex-labeled trees on vertices.
Paper Structure (32 sections, 34 theorems, 98 equations, 6 figures)

This paper contains 32 sections, 34 theorems, 98 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Introduction
  3. The affine symmetric group
  4. Trees
  5. Cyclic trees
  6. Subwords
  7. Subwords and cyclic trees
  8. Enumeration
  9. The affine symmetric group
  10. Trees
  11. Tree-like factorizations
  12. Tree embeddings
  13. Enumeration
  14. Cyclic trees
  15. Cyclic factorizations
  16. ...and 17 more sections

Key Result

Theorem \ref{thm:tree_like}

There is a bijection between $\underline{\textsc{tree}}_n$ and $\underline{\textsc{f}\widetilde{\textsc{a}}\textsc{ct}}_n$, where $\underline{\textsc{tree}}_n$ is the set of plane-embedded vertex-labeled trees on $[n]$ with a marked edge adjacent to the vertex $n$ (up to orientation preserving homeo

Figures (6)

  • Figure 1: Our running example. Left: a cyclically-embedded vertex-labeled tree in $\textsc{tree}_{9}$ (for now, ignore the arrows, green edges, and green labels). Right: the corresponding maximal distinguished subword $\mathsf{u} \in \textsc{sub}_{10}$, with takes in green, and skips in white and purple (decorated by the corresponding skip reflection, with the convention that $\overline{i}\coloneqq i-n$).
  • Figure 2: The $30=3!\cdot \mathrm{Cat}(3)$ trees in $\underline{\textsc{tree}}_4$. Each tree has only the vertex $4$ labeled, and so corresponds to $3!$ vertex-labeled trees in $\underline{\textsc{tree}}_4$ by choosing a labeling of the unlabeled vertices by $1,2,3$.
  • Figure 3: The set $\textsc{tree}_4$, the 16 vertex-labeled trees on $4$ vertices, cyclically embedded in the plane according to \ref{['sec:cyclic_from_trees']}. Below each tree is the corresponding cyclic factorization of $\lambda_4$
  • Figure 4: $\textsc{inv}(\mathsf{u})$ for the maximal distinguished subword from \ref{['fig:main']}. Skips are colored purple.
  • Figure 5: The $16$ distinguished subwords in $\textsc{sub}_4$, with letters chosen in the subword indicated in green, positive skips in white, and negative skips in purple (and replaced by the corresponding inversions). Compare with \ref{['fig:trees']}.
  • ...and 1 more figures

Theorems & Definitions (84)

  • Theorem \ref{thm:tree_like}
  • Corollary \ref{thm:tree_like}
  • Theorem \ref{thm:main_thm}
  • Theorem \ref{thm:subword_count}: P. Galashin, T. Lam, N. Williams
  • Corollary \ref{thm:subword_count}: Cayley's formula
  • Remark \ref{thm:subword_count}
  • Proposition \ref{thm:subword_count}
  • Proposition \ref{thm:subword_count}
  • proof
  • Definition \ref{thm:subword_count}
  • ...and 74 more