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A Colorful Way to Park: An Introduction to Exact $k$-Typed Parking Functions

Aalliyah Celestine, Jacob van der Leeuw, Lina Liu

TL;DR

This paper defines exact $k-typed parking functions ($k$-TPFs) to be a variant of classical parking functions and establishes that every exact $k$-TPF $\alpha$ of length $M$ corresponds to a unique parking configuration $C$.

Abstract

Parking functions are tuples that describe the parking of $M$ cars on a street with $M$ parking spots. In this paper, we define exact $k$-typed parking functions ($k$-TPFs) to be a variant of classical parking functions. We then establish that every exact $k$-TPF $α$ of length $M$, corresponds to a unique parking configuration $C$. We observe that the collection of all exact $k$-TPFs which result in the same configuration form a disjoint subset of all exact $k$-TPFs. Lastly, we conclude by showing how parking permutations of an exact $k$-TPF can be related to other combinatorial objects.

A Colorful Way to Park: An Introduction to Exact $k$-Typed Parking Functions

TL;DR

This paper defines exact kk\alphaMC$.

Abstract

Parking functions are tuples that describe the parking of cars on a street with parking spots. In this paper, we define exact -typed parking functions (-TPFs) to be a variant of classical parking functions. We then establish that every exact -TPF of length , corresponds to a unique parking configuration . We observe that the collection of all exact -TPFs which result in the same configuration form a disjoint subset of all exact -TPFs. Lastly, we conclude by showing how parking permutations of an exact -TPF can be related to other combinatorial objects.
Paper Structure (6 sections, 15 theorems, 24 equations, 2 figures)

This paper contains 6 sections, 15 theorems, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Basic properties of $k$-TPFs
  3. Counting the Number of $k$-typed parking functions
  4. Cumulative Results on Mapping Families to Configurations
  5. Counting groupings of $k$-typed parking functions
  6. Discussion and Future Works

Key Result

lemma 1

Let $M=m_1+m_2$. If $\alpha=(m_1;P_2)$ with $P_2=(p_1^{(2)},p_2^{(2)},\ldots,p_{m_2}^{(2)})$ satisfies $p_j^{(2)}\in\{0,1,2,\ldots, m_1\}$ for all $1\leq j\leq m_2$, then $\alpha \in \mathrm{eTPF}(M,2)$.

Figures (2)

  • Figure 1: Example of parking configuration & an element in fam($\alpha$)
  • Figure 2: Parking cars using "at-least"

Theorems & Definitions (39)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • lemma 1
  • proof
  • lemma 2
  • proof
  • definition 5
  • theorem 1
  • ...and 29 more