Noncrossing Sets & Rational Kirkman Numbers

• The paper introduces noncrossing sets as combinatorial collections defined by admissibility and shellability, linking geometry, algebra, and Catalan theory.
• It employs explicit formulas, partial Springer resolutions, and Deodhar decompositions to enumerate configurations via rational associahedra and Hecke algebra traces.
• The study bridges classical rational Catalan, Kirkman, and parking numbers, revealing deep connections with representation theory and knot invariants.

Noncrossing sets for rational Kirkman numbers serve as a central combinatorial framework bridging the geometry of partial resolutions, representation theory of Hecke algebras, and rational extensions of Catalan and parking function objects. These sets appear both as explicit combinatorial collections—mutually noncrossing configurations defined by admissibility and shellability—and as geometric entities arising from decompositions of varieties attached to Coxeter and braid group data. Their cardinalities are given by the rational Kirkman numbers, which interpolate between classical rational Catalan numbers and rational parking numbers, and are conjecturally the $f$-vectors of rational associahedra.

1. Definition and Algebraic Foundation of Rational Kirkman Numbers

Let $n$ and $p$ be positive integers with $\gcd(p,n)=1$, and set $W=S_n$ of rank $r=n-1$. The standard reflection representation $V$ of $W$ over $\mathbb{Q}$ admits the Galois twist $V_p$ with character $\chi_{V_p}(w)=\chi_V(w^p)$. For $0\le k\le r$, the $k$th rational Kirkman polynomial is: $$\operatorname{Kirk}_{n,p}^{(k)}(X) = \frac{\det(1-X^p e_{\wedge^k}|V_p^*)}{\det(1-X e_{\wedge^k}|V^*)},$$ where $e_{\wedge^k}\in\mathbb{Q}[W]$ is the idempotent projecting onto $\wedge^k V$ (Trinh et al., 24 Jan 2026). The rational Kirkman numbers are the specializations $$\operatorname{Kirk}_{n,p}^{(k)} = \operatorname{Kirk}_{n,p}^{(k)}(1)$$.

In the normalization $p=n+1$, these specialize to classical Kirkman $f$-vectors associated with the associahedron, and in general, they conjecturally yield the $f$-vectors of the “rational associahedron” (Armstrong et al., 2013). These numbers interpolate between rational Catalan numbers ($k=0$) and rational parking numbers ($k=r$), appearing as graded multiplicities of $\wedge^k$ in modules from rational Cherednik algebra theory.

2. Geometric Realization via Partial Springer Resolutions

Partial resolutions of the nilpotent cone in type $A$, specifically Springer resolutions, provide the geometric underpinnings for noncrossing sets enumerated by rational Kirkman numbers. For $G = GL_n(\mathbb{F}_q)$, $B$ its Borel, and subsets $J\subset S$, two partial Springer resolutions are defined: $$\operatorname{Spr}_J^- = \{(u,\,g P_J)\mid u\in gU_Jg^{-1}\},\quad \operatorname{Spr}_J^+ = \{(u,\,g P_J)\mid u\in gV_Jg^{-1}\},$$ the latter corresponding to larger, more interesting cases (Trinh et al., 24 Jan 2026). These resolutions yield central elements in the Iwahori–Hecke algebra $H$ via the Harish-Chandra transform. In type $A$, explicit formulas for these central elements involve summing over distinguished coset representatives $v\in W^{J,\pm}$.

The morphism $\text{mult}: E_J^\pm\to (G/B)^2$ induces elements $\Sigma_{J,\pm}$ in the Hecke algebra, as detailed: $$\text{mult}_!(1_{E_J^-}) = (q-1)^{n-1}\Sigma_{J,-},\qquad \Sigma_{J,-} = \sum_{v\in W^{J,-}} q^{-\ell(v)} T_{v^{-1}} T_v.$$ This geometric route underpins the enumeration and structural characterization of noncrossing sets by connecting variety stratification to Hecke algebra trace evaluations.

3. Combinatorial Realization: Rational Associahedra and Dyck Paths

For coprime $a < b$, the rational associahedron $\operatorname{Ass}(a,b)$ is a pure simplicial complex of dimension $a-2$ constructed by labeling the vertices of a convex $(b+1)$-gon and defining admissible diagonals as those separating sets of vertices corresponding to the sequence $S(a,b) = \{\lfloor jb/a\rfloor: 1\le j\le a-1\}$. Faces of $\operatorname{Ass}(a,b)$ are collections of mutually noncrossing admissible diagonals; its maximal faces are bijective with $(a,b)$-Dyck paths—lattice paths in $\mathbb{Z}^2$ from $(0,0)$ to $(b,a)$ that stay above $y=(a/b)x$ (Armstrong et al., 2013).

A key enumerative result is the product formula for the rational Kirkman numbers (the $k$-dimensional faces of $\operatorname{Ass}(a,b)$): $$f_k(a,b) = \frac{1}{a} \binom{a}{k+2} \binom{b+k+1}{k+1},$$ for $0 \le k \le a-2$. These numbers achieve a uniform combinatorial enumeration of noncrossing sets in the rational context.

4. Deodhar Decompositions and Point-Counting Methodology

Deodhar-style decompositions stratify Richardson and braid varieties associated with Coxeter group word powers. For a $p$-fold Coxeter word $c^p$, distinguished subwords $\omega$ decompose the associated varieties into cells characterized by the numbers of down-steps $d_\omega$ and stays $e_\omega$: $$|R^{(v)}(c^p)_\omega| = q^{|d_\omega|}(q-1)^{|e_\omega|}.$$ Analogous decompositions apply to partial braid–Steinberg varieties $Z_J^+(c^p)$, leading to formulas tallying points in terms of sums over cells, ultimately tied to central elements in the Hecke algebra (Trinh et al., 24 Jan 2026). This stratified enumeration validates the Kirkman numbers via cell counts and connects topological link invariants (HOMFLYPT, Markov traces) to algebraic combinatorics.

5. Construction and Characterization of Noncrossing Sets

Noncrossing sets enumerated by rational Kirkman numbers arise from refined combinatorial or geometric conditions. In the combinatorial context, for $v\in S_n$ with specified left-ascent set $\operatorname{Asc}(v)=I_k$ (with $I_k=\{s_1,\dots,s_{n-1-k}\}$), the set $M^{(v)}(c^p)$ of distinguished subwords with minimal $|e_\omega|=r$ yields uniform noncrossing set constructions across the parameter space $(n,p,k)$: $$\operatorname{Kirk}_{n,p}^{(k)} = \sum_{\operatorname{Asc}(v)=I_k} |M^{(v)}(c^p)|.$$ Geometrically, each $v$ can be written as a product of specific standard Coxeter elements, and the associated subword conditions generalize parking-function data, extending the classical noncrossing parking function sets. Explicit enumeration for small $n$ is available, and full tabulation of noncrossing sets by Kirkman number values is provided in representative cases ((Trinh et al., 24 Jan 2026) Figure 1).

In rational associahedron theory, the homogeneous construction via Dyck paths produces noncrossing set partitions (injective maps from Dyck paths to set partitions $\mu(D)$), guaranteeing noncrossingness and supporting cyclic sieving phenomena (Armstrong et al., 2013).

6. Interpolation Between Catalan, Parking, and Kirkman Objects

The noncrossing sets for rational Kirkman numbers interpolate between several classical combinatorial invariants:

• For $k=0$, the size of the set recovers the rational Catalan numbers, which count maximal noncrossing collections of admissible diagonals.
• For $k=r$, the number coincides with the rational parking numbers, connected to graded multiplicities in Cherednik algebra modules.
• For intermediate $k$, the noncrossing sets combine features of rational parking functions, higher parking sets, and the general $f$-vector entry construction for rational polytopes (Trinh et al., 24 Jan 2026, Armstrong et al., 2013).

This interpolation marks rational Kirkman numbers as the pivotal quantities organizing the full hierarchy of rational noncrossing objects.

7. Connections to Link Invariants, Trace Formulas, and Representation Theory

The enumeration of noncrossing sets for rational Kirkman numbers is closely connected to trace formulas in Hecke algebra theory and topological link invariants. In particular, HOMFLYPT and Markov traces $\mu_n^{(k)}$ on $H_{S_n}$ are given by evaluation-trace formulas involving the same central elements $\zeta_{I_k}^+$ that govern the Deodhar cell enumeration: $$\mu_n^{(k)}(T_{c^p}) = \frac{1}{(X-1)^{n-1}} \sum_{\operatorname{Asc}(v)=I_k}\sum_{\omega\in D^{(v)}(c^p)} X^{|d_\omega|} (X-1)^{|e_\omega|}.$$ A BGG-resolution argument in rational Cherednik theory further identifies these trace evaluations with the rational Kirkman polynomials, establishing algebraic and topological bridges between counting formulas and knot invariants (Trinh et al., 24 Jan 2026).

The synthesis of partial Springer resolution geometry, Hecke algebraic centrality, braid variety decompositions, trace identities, and rational Cherednik module character formulas constitutes the foundation for the explicit and uniform construction of noncrossing sets and the proof of their enumeration by rational Kirkman numbers.