Bigraded Rational Parking Functions

• Bigraded rational parking functions are combinatorial objects defined on rational Dyck paths with dual gradings (area and dinv) that encode complex algebraic and geometric properties.
• They connect symmetric functions, Schur positivity, and Hilbert series expansions with algebraic structures like diagonal coinvariant algebras and Hitchin moduli spaces.
• Applications extend to generalizations such as rational Catalan numbers, shuffle conjectures, and refined enumerative models in modern representation theory.

A bigraded rational parking function is a combinatorial object generalizing classical parking functions to settings indexed by coprime integers $(m,n)$, equipped with two gradings derived from key statistics (most notably "area" and "dinv") on rational Dyck paths or related tableaux. These bigraded objects encode deep connections between algebraic combinatorics, symmetric functions, and geometric representation theory, with their graded enumeration governed by polynomials exhibiting Schur positivity, symmetric function expansions, and geometric interpretations via the cohomology of moduli spaces such as wild Hitchin systems.

1. Combinatorial Definition and Structure

A rational parking function of type $(m,n)$ (for coprime $m,n$) is a labeling of the north steps of a rational $(m,n)$-Dyck path: a lattice path from $(0,0)$ to $(n,m)$ using $m$ north and $n$ east steps that never crosses below the diagonal $y=(m/n)x$. The north steps are labeled by distinct integers $1,\ldots,m$, with the constraint that labels in each column are strictly increasing.

Bigraded rational parking functions are these parking functions endowed with two statistics:

• area$(P)$: counts the number of boxes between the Dyck path and the diagonal inside the $n \times m$ rectangle.
• dinv$(P)$: a generalized diagonal inversion statistic reflecting affine permutation inversions or tableau content.

Formally, the set of bigraded rational parking functions $PF_{m,n}$ can be described via affine permutations or invariant sets, as detailed in (Nelson, 28 Feb 2025, Gorsky et al., 2014), and (Armstrong et al., 2014). The statistics $(\operatorname{area}, \operatorname{dinv})$ define a double grading on $PF_{m,n}$.

2. Symmetric Function and Hilbert Series Expansion

Bigraded enumeration is achieved via the Hilbert series or Frobenius series: $$PF_{m,n}(q,t) = \sum_{P \in PF_{m,n}} q^{\operatorname{dinv}(P)}\, t^{\operatorname{area}(P)}\, F_{n,IDes(P)}(x)$$ where $F_{n,IDes(P)}(x)$ is a fundamental quasisymmetric function determined by the inverse descent set of the labeling permutation.

For coprime $(m,n)$, this polynomial admits a Schur expansion: $$PF_{m,n}(q,t) = \sum_{\lambda \vdash n} f_\lambda(q,t)\, s_\lambda(x)$$ where $s_\lambda(x)$ is the Schur function and $f_\lambda(q,t)$ are nonnegative Laurent polynomials recording graded multiplicities (see (Xie, 25 Sep 2025, Armstrong et al., 2014, Gorsky et al., 2015)).

Key properties:

• Schur positivity: $PF_{m,n}(q,t)$ is symmetric and Schur-positive (Gorsky et al., 2015).
• Rational Catalan specialization: Setting $q = t = 1$ yields enumeration by the rational Catalan number $Cat_{m,n} = \frac{1}{n}\binom{m+n}{n}$ (Armstrong et al., 2014).
• Hook expansion: Coefficients of hook-shaped Schur functions correspond to rational Schröder numbers.

3. Algebraic and Geometric Interpretation

The bigraded rational parking function polynomial $PF_{m,n}(q, t)$ encodes the structure of the diagonal coinvariant algebra and characterizes the bigraded cohomology of objects such as the wild Hitchin moduli space for type $A_{n-1}$ with prescribed singularity. For a "full regular" singularity, $PF_{m,n}(q,t)$ equals the bigraded polynomial $C_{m,n}(q,t)$ computing the mixed Hodge polynomial or perverse filtration polynomial of the Hitchin system (Xie, 25 Sep 2025). For general parabolic type $f$, the polynomial generalizes to

$$C^f_{m,n}(q,t) = \sum_{\lambda} f_\lambda(q,t) K_{\lambda f}$$

where $K_{\lambda f}$ are Kostka numbers reflecting multiplicity imposed by the type of singularity.

Affine Springer fiber geometry gives a cell decomposition matching graded parameters (codimension and combinatorial statistics), and the Anderson map relates points in affine Weyl group orbits to bigraded parking functions (Gorsky et al., 2014, Gorsky et al., 2015).

4. Representation Theory and Rational Shuffle Conjecture

The rational shuffle conjecture predicts that the bigraded Frobenius characteristic of rational parking functions coincides with the graded character of diagonal harmonics or Cherednik algebra representations: $$P_{m,n} \cdot 1 = \sum_{f \in PF_{m/n}} q^{\operatorname{area}(f)}\, t^{\operatorname{dinv}(f)}\, Q_{ides(f)}(z)$$ where $Q_{ides(f)}(z)$ is the Gessel fundamental quasisymmetric function associated to the inverse descent set (Gorsky et al., 2015). Each fixed rational Dyck path contributes a skew LLT polynomial, and the associated statistics (area, dinv, ides) align with ribbon tableau content and spin via the rational Anderson bijection.

Symmetric group $S_n$ acts naturally on $PF_{m,n}$, and the orbit structure matches Schur expansions and the rational Catalan combinatorics (Armstrong et al., 2014).

5. Explicit Enumeration, q,t-analogues, and Nonstandard Models

The polynomial $PF_{m,n}(q,t)$ generalizes classical $q,t$-Catalan numbers and admits refined combinatorial interpretations:

• q-analogue: weights via area or dinv to recover rational $q$-Catalan numbers;
• t-analogue: parameterizes secondary statistics (e.g., bounce, via sweep maps);
• joint symmetry: conjectured $PF_{m,n}(q,t) = PF_{m,n}(t,q)$ (Armstrong et al., 2014);
• sweep map: bijective methods generalizing zeta maps exchange area and dinv for rational Dyck paths (Armstrong et al., 2014, McCammond et al., 2019, Nelson, 28 Feb 2025).

Novel combinatorial models employ statistics on partitions inside rectangles, nonstandard arm-leg counts along frontiers, and multi-shuffle decompositions (see (Yin, 2021, Yin, 2021, Ehrenborg et al., 2017)).

6. Generalizations and Structural Extensions

Variants include $(r,k)$-parking functions, vector parking functions, and interval parking functions. Many such generalizations inherit a natural bigrading when additional statistics (e.g., displacement, boundary parameters) are considered (Stanley et al., 2016, Yang, 8 May 2024, Aguilar-Fraga et al., 2023). Hopf algebraic frameworks reflect these gradings (see (Li, 2015)), and combinatorial decompositions—such as block structures in Fubini rankings—suggest refined bigradings mapped to symmetric function bases (Bradt et al., 13 Jan 2024).

Applications extend to the paper of grossman-larson Hopf algebras, moduli of tiered graphs, spanning trees, and connections to Postnikov-Shapiro algebras (Bagchi et al., 6 Aug 2024).

7. Connections to Moduli Spaces, Perverse and Weight Filtrations

The combinatorially defined bigraded rational parking function polynomial is conjectured to agree with geometrically defined bigraded polynomials arising from the perverse filtration of the Hitchin fibration or the weight filtration on the mixed Hodge structure of the character variety (Xie, 25 Sep 2025). Affine Springer fiber decomposition, Anderson's bijection, and geometric constructions yield cell dimensions matching combinatorial gradings, and Schur expansions encapsulate representation-theoretic information.

A summary of key polynomials:

Polynomial	Role/Interpretation	Main Formula(s)
$PF_{m,n}(q,t)$	Bigraded rational parking function generating function	$$\sum_{P} q^{\operatorname{dinv}(P)} t^{\operatorname{area}(P)} F_{n,IDes(P)}(x)$$
$C_{m,n}(q,t)$	Cohomology polynomial for Hitchin system (full regular)	$\sum_\lambda f_\lambda(q,t) s_\lambda(x)$
$C^{f}_{m,n}(q,t)$	Cohomology polynomial with parabolic type $f$	$\sum_\lambda f_\lambda(q,t) K_{\lambda f}$

In conclusion, bigraded rational parking functions integrate combinatorial, algebraic, and geometric structures via double gradings derived from Dyck path statistics. Their enumeration via symmetric function expansions reflects deep links with the cohomology of moduli spaces, rational Catalan and Schröder numbers, and modern representation theory.