arXiv subjects

Weikang Liang

Publications and source records attributed to Weikang Liang.

7 recordsLinked to original sources

arXiv2026

Freeness of Arrangements with Regular Underlying Matroids

We classify freeness for finite central arrangements whose underlying matroids are regular. Let $\mathcal A$ be such an arrangement over an arbitrary field, and put $M=M(\mathcal A)$. Then $\mathcal A$ is free if and only if $M$ is supersolvable; equivalently, $M$ admits a nice partition; equivalently, $M$ is the cycle matroid of a chordal simple graph. Thus, for arrangements with regular underlying matroids, freeness has a complete combinatorial classification independent of the base field. We use Seymour's decomposition theorem for regular matroids to prove that freeness forces supersolvability. We also characterize nice partitions of finite simple binary matroids: a partition is nice if and only if it is independent and no line is contained in a single block. Consequently, a finite loopless binary matroid admits a nice partition if and only if it is simple and supersolvable.

math.CO↗

arXiv2026

Level of Faces for Exponential Sequence of Arrangements

In this paper, we introduce the bivariate exponential generating function $F_l(x,y)$ for the number of level-$l$ faces of an exponential sequence of arrangements (ESA), and establish the formula $F_l(x,y)=\big(F_1(x,y)\big)^l$ with a combinatorial interpretation. Its specialization at $x=0$ recovers a result first obtained by Chen et al. [3,4] for certain classic ESAs and later generalized to all ESAs by Southerland et al. [8]. As a byproduct, we obtain that an alternating sum of the number of level-$l$ faces is invariant with respect to the choice of ESA, and is exactly the Stirling number of the second kind. We also extend the binomial-basis expansion theorem [3,4,14] and Stanley's formula on ESAs [9] from characteristic polynomials to Whitney polynomials.

math.CO↗

arXiv2025

Several New Generalizations of LYM Inequality

The LYM inequality is a fundamental result concerning the sizes of subsets in a Sperner family. Subsequent studies on the LYM inequality have been generalized to families of $r$-decompositions, where all components are required to avoid chains of the same length. In this paper, we relax this constraint by allowing components of a family of $r$-decompositions to avoid chains of distinct lengths, and derive generalized LYM inequalities across all the relevant settings, including set-theoretic, $q$-analog, continuous analog, and arithmetic analog frameworks. Notably, the bound in our LYM inequalities does not depend on the maximal length of all forbidden chains. Moreover, we extend our approach beyond $r$-decompositions to $r$-multichains, and establish analogous LYM inequalities.

math.CO↗

arXiv2025

A decomposition of Grassmannian associated with a hyperplane arrangement

The Grassmannian, which is the manifold of all $k$-dimensional subspaces in the Euclidean space $\mathbb{R}^n$, was decomposed through three equivalent methods connecting combinatorial geometries, Schubert cells and convex polyhedra by Gelfand, Goresky, MacPherson and Serganova. Recently, Liang, Wang and Zhao discovered a novel decomposition of the Grassmannian via an essential hyperplane arrangement, which generalizes the first two methods. However, their work was confined to essential hyperplane arrangements. Motivated by their research, we extend their results to a general hyperplane arrangement $\mathcal{A}$, and demonstrate that the $\mathcal{A}$-matroid, the $\mathcal{A}$-adjoint and the refined $\mathcal{A}$-Schubert decompositions of the Grassmannian are consistent. As a byproduct, we provide a classification for $k$-restrictions of $\mathcal{A}$ related to all $k$-subspaces through two equivalent methods: the $\mathcal{A}$-matroid decomposition and the $\mathcal{A}$-adjoint decomposition.

math.CO↗

arXiv2025

Modular Ideals and Factorizations of Affine Hyperplane Arrangements

Let $\mathcal{A}$ be an affine hyperplane arrangement, let $L(\mathcal{A})$ be its intersection semilattice, and let $\chi_{\mathcal{A}}(t)$ be its characteristic polynomial. We introduce ideal decompositions of finite ranked meet-semilattices and prove that they induce purely combinatorial factorizations of characteristic polynomials. This framework recovers both Stanley's factorization associated with modular elements and Terao's factorization associated with nice partitions. For simple semimatroids, every join-closed ideal decomposition comes from a direct-sum decomposition. The two-factor case defines modular ideals, extending the role of modular elements from central to affine arrangements. Over an infinite field, every modular ideal of an arrangement intersection semilattice is realized by a suitable translated restriction of a subarrangement. We exhibit a nonempty Zariski-open set of valid translations, yielding essential realizations of the given modular ideal. We further prove that modular subarrangements correspond under coning to modular elements on the hyperplane at infinity and hence to M-ideals in the affine setting. Finally, every modular ideal gives an Orlik-Solomon graded vector-space decomposition, which becomes a graded-algebra decomposition when the two factors come from subarrangements.

math.CO↗

arXiv2025

$k$-Adjoint of Hyperplane Arrangements

In this paper, we introduce the $k$-adjoint of a given hyperplane arrangement $\mathcal{A}$ associated with rank-$k$ elements in the intersection lattice $L(\mathcal{A})$, which generalizes the classical adjoint proposed by Bixby and Coullard. The $k$-adjoint of $\mathcal{A}$ induces a decomposition of the Grassmannian, which we call the $\mathcal{A}$-adjoint decomposition. Inspired by the work of Gelfand, Goresky, MacPherson, and Serganova, we generalize the matroid decomposition and refined Schubert decomposition of the Grassmannian from the perspective of $\mathcal{A}$. Furthermore, we prove that these three decompositions are exactly the same decomposition. A notable application involves providing a combinatorial classification of all the $k$-dimensional restrictions of $\mathcal{A}$. Consequently, we establish the anti-monotonicity property of some combinatorial invariants, such as Whitney numbers of the first kind and the independece numbers.

math.CO↗

arXiv2024

Characterizing Nice Partition of Graphical Arrangements

The successive works of Terao as well as Stanley revealed that, for graphical arrangements, supersolvability and the existence of nice partitions are equivalent properties, both characterized by chordal graphs. In this paper, we further prove that every nice partition of a graphical arrangement arises precisely from a maximal modular chain in its intersection lattice. Moreover, we establish two converses to classical results of Orlik and Terao on nice partitions.

math.CO↗