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Luck and magic for Pitman-Stanley polytopes and parking functions

Nicolas Avila, Luis Ferroni, Alejandro H. Morales

Abstract

Motivated by the combinatorics of parking functions and their several generalizations, we study the Ehrhart theory of Pitman--Stanley polytopes. We prove a strong positivity phenomenon called \emph{magic positivity} for the Ehrhart polynomials of these polytopes, which in turn implies that their $h^*$-polynomials are real-rooted (and thus log-concave and unimodal). Our result is achieved by interpreting the coefficients of these Ehrhart polynomials in the \emph{magic basis} in terms of the number of \emph{lucky cars} in a modified parking protocol. Furthermore, we address the magic positivity problem for $\mathbf{y}$-generalized permutohedra and also discuss a \emph{magic} combinatorial interpretation for them, under the assumption that the input parameters are sufficiently large.

Luck and magic for Pitman-Stanley polytopes and parking functions

Abstract

Motivated by the combinatorics of parking functions and their several generalizations, we study the Ehrhart theory of Pitman--Stanley polytopes. We prove a strong positivity phenomenon called \emph{magic positivity} for the Ehrhart polynomials of these polytopes, which in turn implies that their -polynomials are real-rooted (and thus log-concave and unimodal). Our result is achieved by interpreting the coefficients of these Ehrhart polynomials in the \emph{magic basis} in terms of the number of \emph{lucky cars} in a modified parking protocol. Furthermore, we address the magic positivity problem for -generalized permutohedra and also discuss a \emph{magic} combinatorial interpretation for them, under the assumption that the input parameters are sufficiently large.
Paper Structure (23 sections, 36 theorems, 114 equations, 4 figures, 2 tables, 1 algorithm)

This paper contains 23 sections, 36 theorems, 114 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Overview
  3. Background and notation
  4. --Extended permutations
  5. y-parking functions
  6. Pitman--Stanley polytopes and generalized permutohedra
  7. Real-rootedness and magic basis
  8. Magic positivity and lucky statistic on --extended permutations
  9. First steps into magic positivity
  10. Algebraic contributions
  11. Block parking protocol
  12. Applying the machinery to generalized permutohedra
  13. Magic positivity and lucky statistic on --Parking Functions
  14. Generalized Parking Functions
  15. Lucky statistic on --Parking Functions
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\Pi_n(\mathbf{y})$ be a Pitman--Stanley polytope for $\mathbf{y} \in \mathbb{Z}^n_{>0}$. Let Then, for each $0\leq i \leq n$, the number $n!\cdot c_i$ enumerates the $\mathbf{y}$-parking functions having exactly $i$ lucky cars without the first available space. In particular, each $c_i$ is a nonnegative number, and therefore magic positivity holds.

Figures (4)

  • Figure 1: Example of the block parking protocol: the number on the car is its index, the number inside the black box is the parking space, and the number of the purple box is its initial preference.
  • Figure 2: Example of block parking protocol with one unavailable space.
  • Figure 3: Example of parking protocol with the first space unavailable.
  • Figure 4: The parking protocol is illustrated for each $\mathbf{y}$-parking function $\sigma_i \in \operatorname{PF}_2(2,1)$, where the green cars are lucky cars.

Theorems & Definitions (88)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: $\mathbf{y}$-extended permutations
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Remark 2.5
  • Definition 2.6: $\mathbf{y}$-parking functions
  • Remark 2.7
  • ...and 78 more