Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Weary Drivers, Records of Trees, and Parking Functions

Adrián Lillo, Mercedes Rosas, Stefan Trandafir

TL;DR

This work develops a record-based decomposition of Cayley trees and a blob-like record code, then builds a wearied-parking framework that bijectively links Cayley trees to parking functions while preserving records. It introduces the weary parking process, the priority and arrival trees, and the parking tree, culminating in a record-preserving bijection that explains an equidistribution of a hexad of statistics between Cayley trees and parking functions. The results are supported by generating-function identities and extended through notions of record duality, with concrete computational implementations in Sage. Overall, the paper provides a structural bridge between tree- and parking-function combinatorics and demonstrates powerful statistic-preserving correspondences.

Abstract

This work builds on a decomposition of a Cayley tree using the notion of a record, a concept closely related to the blob encoding of a tree introduced by Kreweras and Moszkowski, and Picciotto, that we explore and extend. We provide an alternative definition of parking functions and derive from it a record-preserving bijection between Cayley trees and parking functions. Finally, we use this bijection to establish an equidistribution between a sextuple of statistics on Cayley trees and a corresponding sextuple of statistics on parking functions.

On Weary Drivers, Records of Trees, and Parking Functions

TL;DR

This work develops a record-based decomposition of Cayley trees and a blob-like record code, then builds a wearied-parking framework that bijectively links Cayley trees to parking functions while preserving records. It introduces the weary parking process, the priority and arrival trees, and the parking tree, culminating in a record-preserving bijection that explains an equidistribution of a hexad of statistics between Cayley trees and parking functions. The results are supported by generating-function identities and extended through notions of record duality, with concrete computational implementations in Sage. Overall, the paper provides a structural bridge between tree- and parking-function combinatorics and demonstrates powerful statistic-preserving correspondences.

Abstract

This work builds on a decomposition of a Cayley tree using the notion of a record, a concept closely related to the blob encoding of a tree introduced by Kreweras and Moszkowski, and Picciotto, that we explore and extend. We provide an alternative definition of parking functions and derive from it a record-preserving bijection between Cayley trees and parking functions. Finally, we use this bijection to establish an equidistribution between a sextuple of statistics on Cayley trees and a corresponding sextuple of statistics on parking functions.

Paper Structure

This paper contains 19 sections, 16 theorems, 24 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The record decomposition and the record code of a Cayley tree.
  3. The record decomposition
  4. The record code
  5. Some properties of the record decomposition.
  6. Weary Drivers Park Fast
  7. Weary parking
  8. From Cayley trees to parking functions
  9. The priority vector.
  10. The arrival tree.
  11. The parking tree.
  12. A record preserving bijection between parking functions and Cayley trees.
  13. An equidistribution result between statistics for Cayley trees and parking functions
  14. Statistics on parking functions
  15. Statistics of Cayley trees
  16. ...and 4 more sections

Key Result

Lemma 2.2

Let $T$ be a Cayley tree labelled $[n]_0$, and let $i$ be a node of $T$. If $i=\circ$, the number of children of $i$ is equal to the multiplicity of $i=\circ$ in $\mathop{\mathrm{RecordCode}}\nolimits(T)$ plus one. If $i \neq \circ$, the number of children of $i$ is equal to the multiplicity of $i$

Figures (4)

  • Figure 1: (\ref{['Fig:running_example']}) A Cayley tree $T$ labelled with $[9]_0$ with record nodes $\{5, 7, 8, 9\}$. (\ref{['Fig:record_edges_deletion']}) The tree $T$ with edges beginning in a record node deleted. (\ref{['Fig:record_attachment_seq']}) The bonsai sequence of $T$. Its attachment sequence is $(\circ, 5, 1).$
  • Figure 2: All 16 Cayley trees labelled with $[3]_0$ together with their respective record codes.
  • Figure 3: The weary parking process for the tree $T$ of our running example.
  • Figure 4: The classical parking process for $\pi = { \StrSubstitute{5, 2, 5, 3, 1, 6, 1, 2, 6}{;}{\\}[\pfunc@tmp] \StrSubstitute{\pfunc@tmp}{,}{ }[\pfunc@tmp] \pfunc@tmp }$.

Theorems & Definitions (33)

  • Example 2.1
  • Example 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Example 2.5
  • Example 3.1: Weary parking
  • ...and 23 more