[Paper Review] Parking functions and triangulation of the associahedron

Jean-Louis Loday|ArXiv.org|Oct 18, 2005

Advanced Combinatorial Mathematics10 references4 citations

TL;DR

This paper establishes a natural bijection between the top-dimensional simplices of the permutohedron and parking functions, showing both sets have size $(n+1)^{n-1}$. Using an inductive construction based on shuffles and face decompositions, it provides a minimal triangulation of the associahedron and permutohedron compatible with the weak Bruhat order, realizing these polytopes as geometric realizations of simplicial sets.

ABSTRACT

We show that a minimal triangulation of the associahedron (Stasheff polytope) of dimension n is made of (n+1)^{n-1} simplices. We construct a natural bijection with the set of parking functions from a new interpretation of parking functions in terms of shuffles.

Motivation & Objective

• To construct a minimal triangulation of the associahedron (Stasheff polytope) compatible with the Tamari poset and edge orientations.
• To show that the number of top-dimensional simplices in this triangulation is $(n+1)^{n-1}$, matching the count of parking functions.
• To define a natural bijection between the set of parking functions of length $n$ and the top-dimensional simplices of the associahedron.
• To extend the construction to the permutohedron, proposing an analogous inductive structure for its triangulation.

Proposed method

• Construct the associahedron as the geometric realization of a simplicial set using oriented simplices derived from the Tamari poset.
• Define parking functions via a new shuffle-based inductive decomposition: $PF_n = igcup_{p+q=n-1} igracevert \{1,\ldots,p+1\} \times Sh(p,q) \times PF_p \times PF_q \big\}$.
• Use the weak Bruhat order on $S_n$ to orient edges of the permutohedron and ensure compatibility with the triangulation.
• Build the triangulation of the permutohedron $\mathcal{P}^n$ inductively by coning over the South pole using faces not containing it.
• Define $SM(n,p)$ as the set of ordered surjective maps $[n+1] \to [p+1]$ excluding the maximal one $f_0$, and use it to parametrize faces.
• Establish the recursive formula $ZP_n = \bigcup_{p=0}^{n-1} SM(n,p) \times Sh(p,n-p-1) \times ZP_p \times ZP_{n-p-1}$ for the top-dimensional simplices of $\mathcal{P}^n$.

Experimental results

Research questions

• RQ1 Is there a canonical bijection between the top-dimensional simplices of the associahedron and the set of parking functions of length $n$?
• RQ2 Can the associahedron be minimally triangulated such that the simplices are compatible with the Tamari poset's edge orientations?
• RQ3 Does the number $(n+1)^{n-1}$ count both the number of parking functions and the number of top-dimensional simplices in a minimal triangulation of the associahedron?
• RQ4 Can a similar inductive structure based on shuffles and face decompositions be used to triangulate the permutohedron?
• RQ5 What combinatorial objects satisfy the recursive formula for the permutohedron's top-dimensional simplices, analogous to parking functions?

Key findings

• The number of top-dimensional simplices in a minimal triangulation of the associahedron of dimension $n$ is exactly $(n+1)^{n-1}$, matching the number of parking functions of length $n$.
• A natural bijection is constructed between the set of parking functions $PF_n$ and the top-dimensional simplices of the associahedron using a shuffle-based inductive decomposition.
• The triangulation of the associahedron is compatible with the edge orientations induced by the Tamari poset, making it a geometric realization of a simplicial set.
• The permutohedron $\mathcal{P}^n$ admits a triangulation with $\#ZP_n$ top-dimensional simplices, where the sequence begins $1, 1, 4, 34, 488, 10512, \ldots$ for $n = 0$ to $8$, satisfying the recursive formula involving $SM(n,p)$ and shuffles.
• The recursive structure of the permutohedron's triangulation mirrors that of parking functions, suggesting the existence of a combinatorial analogue to parking functions for the permutohedron.