Coparking Functions in Matroid Theory

• Coparking functions are combinatorial objects defined on cycle systems in a matroid, using local inequality conditions to capture uniquely contributed cycle elements.
• They extend classical G-parking functions by establishing bijections with matroid bases and employing adapted chip-firing algorithms for efficient recognition.
• Their framework resolves instances of Stanley’s conjecture by showing that maximal coparking functions yield pure O-sequences that match the matroid h-vector.

Coparking functions are combinatorial objects introduced to model and paper certain structural properties of matroids, inspired by the classical theory of parking functions, chip-firing, and the problem of realizing the h-vector of matroids as pure O-sequences. Specifically, a coparking function is defined as an integer vector associated to a cycle system on a matroid, subject to a local inequality condition involving uniquely contributed elements of cycles. These objects bridge enumerative combinatorics, algebraic combinatorics, and matroid theory by providing a canonical way to construct pure multicomplexes whose degree vectors recover the matroid h-vector, thus resolving instances of Stanley’s conjecture for matroids that admit cycle systems.

1. Definition and Core Structure

Let $M$ be a matroid of rank $r$ and let $g$ denote the corank ($g = |E| - r$). A cycle system on $M$ is a collection of cycles (unions of circuits) $\mathcal{C} = \{C_1, C_2, \dots, C_g\}$ with a “unique union” property: for every nonempty $\sigma \subseteq [g]$ (where $[g] = \{1, 2, \dots, g\}$), the set

$$\bigstar_{j \in \sigma} C_j = \left\{ e \in \bigcup_{j \in \sigma} C_j \;\Big|\; e \;\text{appears in exactly one}\; C_j \right\}$$

is required to be dependent (i.e., contains a circuit).

A coparking function with respect to $\mathcal{C}$ is a function $a: \mathcal{C} \to \mathbb{N}$, equivalently a vector $a = (a_1, \dots, a_g) \in \mathbb{N}^g$, such that for every nonempty $\sigma \subseteq [g]$, there exists some $i \in \sigma$ satisfying

$a_i < |C_i \cap \bigstar_{j \in \sigma} C_j|.$

The degree of a coparking function is defined as $\deg(a) = a_1 + \cdots + a_g$.

Notably, the collection of coparking functions forms an order ideal of $\mathbb{N}^g$ (i.e., closed under component-wise decrease), and, crucially, is a pure multicomplex: all maximal coparking functions have the same degree.

2. Cycle Systems: Construction and Combinatorial Role

Cycle systems on a matroid generalize the concept of cut-sets in graphs. Formally, a cycle system $\mathcal{C}$ satisfies:

• $g = \text{corank}(M) = |E| - r$,
• Each $C_j$ is a cycle (union of circuits),
• For all nonempty $\sigma \subseteq [g]$, $\bigstar_{j \in \sigma} C_j$ is dependent.

The unique union operator ($\bigstar$) ensures the selection of elements exclusive to the chosen subset, isolating the "uniquely contributed" ground set elements. This construction encodes structural redundancies, mirroring the entanglement of cuts or cycles in graphic or cographic matroids, and is key for extending chip-firing techniques and parking function dualities to matroidal settings.

Cycle systems can be explicitly constructed for various matroids:

• For graphic matroids of cones (i.e., adding a cone vertex to a graph), cycle systems can be formed by taking cycles corresponding to faces containing the cone vertex.
• Every graphic matroid of a $K_{3,3}$-free graph admits a cycle system.

3. Principal Results: Pure O-Sequences and h-Vectors

Stanley’s conjecture posits that the h-vector of any matroid independence complex is a pure O-sequence—that is, equal to the degree vector of a pure multicomplex. The main theorem states:

For any matroid $M$ with cycle system $\mathcal{C}$,

$\operatorname{deg}(P^*(\mathcal{C})) = h(M),$

where $P^*(\mathcal{C})$ is the set of all coparking functions (as an order ideal in $\mathbb{N}^g$), and $h(M)$ is the h-vector of $M$. Further, all maximal coparking functions have degree $r(M)-\beta$ where $\beta$ is the number of bridges in $M$.

This result substantiates Stanley’s conjecture for a new, broad class of matroids—including all graphic matroids of cones and $K_{3,3}$-free graphs—by constructing explicit pure multicomplexes modeled by coparking functions.

4. Algorithmic and Bijectional Aspects

A variation of Dhar’s burning algorithm from chip-firing theory is adapted to provide a polynomial-time recognition process for coparking functions. The algorithm proceeds by iteratively removing an index $i$ such that the critical inequality is met for the current subset, "burning" the index; coparking is certified if all are burned.

Moreover, the authors construct explicit bijections between:

• maximal coparking functions and bases of $M$,
• coparking functions of the deletion or contraction of a non-bridge, non-loop element $e\in E$ and those for $M$ itself, via degree-preserving injections.

This framework mirrors and generalizes classical G-parking function theory for graphs, extending the powerful enumeration and structural results from the graphical to the full matroidal context.

5. Deletion, Contraction, and Stability of Coparking Functions

Coparking functions exhibit favorable behavior under standard matroid operations:

• Deletion: For an element $e$ (not a bridge or loop), there is a degree-preserving injection from coparking functions of $M \setminus e$.
• Contraction: There is an analogous map for $M / e$.
• Maximal Degree: All maximal coparking functions maintain identical degree across such simplified minors.

This ensures that induction and recursive arguments, familiar in the computation of the Tutte polynomial and h-vector, can be naturally lifted to the coparking function framework.

6. Connections to Parking Functions, Chip-Firing, and Algebraic Invariants

Coparking functions generalize G-parking functions, foundational in chip-firing and the paper of sandpile groups. For instance:

• In the graphical (or cographic) case, the cycle system is determined by cuts associated to vertices, and coparking functions encode stable chip configurations.
• The correspondence to bases—explicitly, spanning trees—of the matroid underscores their enumerative significance.

Additionally, since the h-vector can be realized as an evaluation of the Tutte polynomial (e.g., $T_M(1, y)$), the coparking function model provides new combinatorial and algebraic insights into matroid invariants.

The relation between these combinatorial structures and the algebraic Cohen–Macaulay property of independence complexes is deepened by this coparking function approach, reinforcing the interface between enumerative combinatorics and commutative algebra.

7. Broader Implications and Further Directions

The introduction of coparking functions opens avenues for generalizing chip-firing, sandpile theory, and the dual combinatorics of parking functions from graphs to abstract matroids. By resolving Stanley’s conjecture for all matroids with a cycle system, the coparking function approach provides a unifying framework for:

• Constructing pure O-sequences,
• Explaining structural similarities between different classes of matroids,
• Developing algorithmic tools for recognition and enumeration.

Their potential to encode bijections with matroid bases and their compatibility with deletion/contraction suggest further combinatorial and algebraic applications, as well as connections to possible dualities in broader variants of parking functions on graphs and matroids.