arXiv subjects

Suijie Wang

Publications and source records attributed to Suijie Wang.

At least 19 recordsLinked to original sources

arXiv2026

The Spherical Hadwiger Theorem

We prove the spherical Hadwiger classification in every dimension. For $n\geq1$, every continuous $SO(n+1)$-invariant valuation on the space of all closed spherical convex sets in $\mathbb{S}^n$ can be written uniquely as a linear combination of the spherical intrinsic volumes $V_0,\ldots,V_n$. The same classification holds when the domain is restricted to sets contained in an open hemisphere. The proof is inductive: it reduces the problem to a vanishing statement for simple valuations and uses a continuous alternating cocycle on tuples of spherical points together with a signed coning transform. Through the cone--sphere correspondence, this yields, for $d\geq2$, the corresponding classification of continuous $SO(d)$-invariant conic valuations on all closed convex cones in $\mathbb{R}^d$, without imposing normalization at the zero cone. In particular, $SO$-invariance implies $O$-invariance in both settings.

math.MG↗

arXiv2026

A Proof of Füredi's Conjecture

We prove Füredi's conjecture on strong Bollobás $t$-systems. For every nonnegative integer $t$, a family of $m\ge2$ pairs of finite sets $(A_i,B_i)$ satisfying $|A_i\cap B_i|\le t$ and $|A_i\cap B_j|>t$ for all $i\ne j$ satisfies \[ \sum_i\binom{|A_i|+|B_i|-2t}{|A_i|-t}^{-1}\le1. \] We first prove a weighted inequality for subspace pairs satisfying the symmetric cross-intersection condition over any field. It follows from a local inequality for graded exterior ideals, proved by induction with projections and colon ideals.

math.CO↗

arXiv2026

Cyclically Compatible Deformations of the Braid Arrangement

We prove a characteristic-polynomial shift formula for two-sided extensions of cyclically compatible deformations of the braid arrangement. For a nonnegative integer matrix $M=(m_{ij})$ with zero diagonal, let $\mathcal{A}_M$ be the arrangement \[ x_i-x_j=s,\qquad 1\le i<j\le n,\quad s\in[-m_{ij},m_{ji}]_{\mathbb{Z}}. \] Given $α,β\in\mathbb{N}^n$, define its two-sided extension $\mathcal{A}_M(α,β)$ by replacing this interval with \[ [-m_{ij}-α_i-β_j,\, m_{ji}+α_j+β_i]_{\mathbb{Z}}. \] Call $M$ cyclically compatible if all pairwise distinct $a,b,c$ with $1\le a,b,c\le n$ satisfy \[ m_{ac}\le m_{ab}+m_{bc}+1. \] Under this condition, for the reduced characteristic polynomial $\widetildeχ(\mathcal{A},t) =\dfrac{χ(\mathcal{A},t)}{t}$, we have \[ \widetildeχ(\mathcal{A}_M(α,β),t) = \widetildeχ(\mathcal{A}_M,t-|α|-|β|). \] The proof uses the finite-field method and a cyclic-gap enumeration formula. We also establish redistribution invariance, study weak-sum perturbations, and give applications to Shi, uniform interval, graphical, and Ferrers-type deformations.

math.CO↗

arXiv2026

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 $χ_{\mathcal{A}}(t)$ 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↗

arXiv2026

Covering Hypercube $mB^n$

The Alon--Füredi theorem determines the minimum number of hyperplanes needed to cover every nonzero vertex of the Boolean cube while avoiding the origin. We study hyperplane coverings with multiplicity for the generalized hypercube $mB^n=\{0,1,\ldots,m\}^n$. Our main result is a polynomial multiplicity theorem extending the theorem of Sauermann and Wigderson to $mB^n$, with sharp degree bounds for polynomials having prescribed high-order zeros. We apply this result to obtain lower and upper bounds for hyperplane coverings with multiplicity and determine the exact minimum number of hyperplanes for coverings with multiplicities $k=1,2$. In the special case of the Boolean cube, our constructions improve the upper bound of Clifton and Huang for certain ranges of the parameters.

math.CO↗

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 Decompositions for Symmetric Deformations of the Braid Arrangement

Let $A\subseteq\mathbb R_{\ge0}$ be finite and nonempty, and let $\mathfrak{A}^A=(\mathcal{A}_1^A,\mathcal{A}_2^A,\ldots)$ be the associated sequence of symmetric deformations of the braid arrangement. Denote by $r_\ell(\mathcal{A}_n^A)$ the number of its level-$\ell$ regions and by $F_\ell(\mathfrak{A}^A,x)$ the corresponding exponential generating function. We prove $F_\ell(\mathfrak{A}^A,x)=\bigl(F_1(\mathfrak{A}^A,x)\bigr)^\ell$. As a consequence, the characteristic polynomial has the binomial-basis expansion $χ(\mathcal{A}_n^A,t)=\sum_{\ell=1}^{n}(-1)^{n-\ell}\,r_\ell(\mathcal{A}_n^A)\binom{t}{\ell}$. When $0\in A$ and $A^*=A\setminus\{0\}$ is nonempty, we refine a classical identity of Stanley level by level: $F_\ell(\mathfrak{A}^{A^*},x)=F_\ell(\mathfrak{A}^{A},1-e^{-x})$. Equivalently, the Catalan-type and semiorder-type level counts satisfy an unsigned Stirling convolution of the first kind. For the $m$-Catalan arrangement $\mathcal{A}_n^{[0,m]}$, we obtain $r_\ell(\mathcal{A}_n^{[0,m]})=n!\,\operatorname{Ran}_{m+1,m\ell}(n-\ell)$, where $\operatorname{Ran}_{p,r}(q)$ is a Raney number. This realizes Raney numbers as refined region counts and answers a question of Deshpande, Menon, and Sarkar. The proofs use labeled Dyck paths, interval orders, and exponential sequences of arrangements. We also realize the inverse Fu--Wang--Zhu bijection for $m$-Catalan regions by tableaux.

math.CO↗

arXiv2026

Log-Concavity of Conic Intrinsic Volumes

Let $n\ge1$ and $C\subseteq\mathbb{R}^n$ be a closed convex cone with conic intrinsic volumes $v_0(C),\ldots,v_n(C)$. We prove the long-standing log-concavity conjecture for this sequence, in the stronger form \[ v_k(C)^2\ge ρ_kρ_{n-k}v_{k-1}(C)v_{k+1}(C), \qquad 1\le k\le n-1, \] where, for $l\ge1$, $ρ_l=\frac{l+1}{l}\frac{ω_{l-1}ω_{l+1}}{ω_l^2}>1$ and $ω_j$ is the volume of the Euclidean unit ball in $\mathbb{R}^j$. The proof applies the Alexandrov--Fenchel inequality to the Takemura--Kuriki identity \[ V(A[k],D[n-k])=\frac{ω_kω_{n-k}}{\binom nk}v_k(C),\qquad A=C\cap B^n,\quad D=C^\circ\cap B^n, \] where $C^\circ$ is the polar cone, $B^n$ is the Euclidean unit ball, and repeated arguments are indicated by brackets. A Master Steiner argument gives the identity directly for arbitrary closed convex cones, including degenerate ones. We also give a circular-cone counterexample to the standard ultra-log-concavity normalizations.

math.CO↗

arXiv2026

Characteristic Polynomials of Graph- and Digraph-Deleted Catalan Arrangements

We develop a finite-field stratification for characteristic polynomials of deletion subarrangements of the full $m$-Catalan arrangement. It reduces the count to cyclic placements of rigid blocks and yields falling-factorial expansions for deletions indexed by graphs, digraphs, and gain-labeled digraphs. The coefficients are graphical Stirling numbers for zero-layer deletions, directed matching numbers when the deleted layer $\ell$ satisfies $1\le \ell\le\lfloor m/2\rfloor$, directed path-cover numbers when $\lfloor m/2\rfloor<\ell\le m$, and admissible gain-labeled arc sets for multilayer deletions. For $\ell=m$, a complementary path-cover expansion yields factorization consequences. The method also gives formulas for directed Ish-type arrangements in terms of path-cycle covers and outdegrees.

math.CO↗

arXiv2026

Convolution-type Identity for Characteristic Polynomials of Geometric Semilattices

We establish a convolution formula for the characteristic polynomial of a finite geometric semilattice $M$: \[ χ(M,st)=\sum_{X\in \underline{M}} s^{r-{\rm rk}_{\underline{M}}(X)}χ(\underline{M}^X,t)\,χ(M_{(X)},s), \] where $\underline{M}$ denotes the centralization of $M$, and $M_{(X)}$ denotes the localization at $X$. This generalizes a nice formula of Southerland, Southern, and Zhou, which is recovered at $s=1$. When specialized to hyperplane arrangements, the identity yields a new expansion closely related to Wang's convolution formula. We further provide a combinatorial interpretation of the convolution formula using the finite field method over $\mathbb{F}_{p^2}$ and $\mathbb{F}_p$.

math.CO↗

arXiv2026

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↗

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

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↗

arXiv2025

The symmetric strong circuit elimination property

If $C_1$ and $C_2$ are circuits in a matroid $M$ with $e_1$ in $C_1-C_2$ and $e$ in $C_1\cap C_2$, then $M$ has a circuit $C_3$ such that $e\in C_3\subseteq (C_1\cup C_2)-e$. This strong circuit elimination axiom is inherently asymmetric. A matroid $M$ has the symmetric strong circuit elimination property (SSCE) if, when the above conditions hold and $e_2\in C_2-C_1$, there is a circuit $C_3'$ with $\{e_1,e_2\}\subseteq C_3'\subseteq (C_1\cup C_2)-e$. We prove that a connected matroid has this property if and only if it has no two skew circuits. We also characterize such matroids in terms of forbidden series minors, and we give a new matroid axiom system that is built around a modification of SSCE.

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

$k$-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

Level of Regions for Deformed Braid Arrangements

This paper primarily investigates a specific type of deformation of the braid arrangement $\mathcal{B}_n$ in $\mathbb{R}^n$, denoted by $\mathcal{B}_n^A$ and defined in (1.2). Let $r_l(\mathcal{B}_n^A)$ be the number of regions of level $l$ in $\mathcal{B}_n^A$ with the corresponding exponential generating function $R_l(A;x)$. Using the weighted digraph model introduced by Hetyei [11], we establish a bijection between regions of level $l$ in $\mathcal{B}_n^A$ and valid $m$-acyclic weighted digraphs on the vertex set $[n]$ with exactly $l$ strong components. Based on this bijection, we obtain a property analogous to a polynomial sequence of binomial type, that is, $R_l(A;x)$ satisfies the relation \[ R_l(A;x)=\big(R_1(A;x)\big)^l=R_k(A;x)R_{l-k}(A;x). \] Furthermore, the values $r_l(\mathcal{B}_n^A)$ yield a combinatorial interpretation for the coefficients in the expansion of the characteristic polynomial $χ_{\mathcal{B}_n^A}(t)$ in the basis elements $\binom{t}{l}$, that is, \[χ_{\mathcal{B}_n^A}(t)=\sum_{l=0}^n(-1)^{n-l}r_l(\mathcal{B}_n^A)\binom{t}{l}.\] If $n$, $a$ and $b$ are non-negative integers with $n\ge 2$ and $b-a\ge n-1$, for the deformation $\mathcal{B}_n^{[-a,b]}$ defined in (1.3), its characteristic polynomial has a single real root $0$ of multiplicity one when $n$ is odd, and has one more real root $\frac{n(a+b+1)}{2}$ of multiplicity one when $n$ is even.

math.CO↗

arXiv2024

Counting Flows of $b$-compatible Graphs

Kochol introduced the assigning polynomial $F(G,α;k)$ to count nowhere-zero $(A,b)$-flows of a graph $G$, where $A$ is a finite Abelian group and $α$ is a $\{0,1\}$-assigning from a family $Λ(G)$ of certain nonempty vertex subsets of $G$ to $\{0,1\}$. We introduce the concepts of $b$-compatible graph and $b$-compatible broken bond to give an explicit formula for the assigning polynomials and to examine their coefficients. More specifically, for a function $b:V(G)\to A$, let $α_{G,b}$ be a $\{0,1\}$-assigning of $G$ such that for each $X\inΛ(G)$, $α_{G,b}(X)=0$ if and only if $\sum_{v\in X}b(v)=0$. We show that for any $\{0,1\}$-assigning $α$ of $G$, if there exists a function $b:V(G)\to A$ such that $G$ is $b$-compatible and $α=α_{G,b}$, then the assigning polynomial $F(G,α;k)$ has the $b$-compatible spanning subgraph expansion \[ F(G,α;k)=\sum_{\substack{S\subseteq E(G),\\G-S\mbox{ is $b$-compatible}}}(-1)^{|S|}k^{m(G-S)}, \] and is the following form $F(G,α;k)=\sum_{i=0}^{m(G)}(-1)^ia_i(G,α)k^{m(G)-i}$, where each $a_i(G,α)$ is the number of subsets $S$ of $E(G)$ having $i$ edges such that $G-S$ is $b$-compatible and $S$ contains no $b$-compatible broken bonds with respect to a total order on $E(G)$. Applying the counting interpretation, we also obtain unified comparison relations for the signless coefficients of assigning polynomials. Namely, for any $\{0,1\}$-assignings $α,α'$ of $G$, if there exist functions $b:V(G)\to A$ and $b':V(G)\to A'$ such that $G$ is both $b$-compatible and $b'$-compatible, $α=α_{G,b}$, $α'=α_{G,b'}$ and $α(X)\leα'(X)$ for all $X\inΛ(G)$, then \[ a_i(G,α)\le a_i(G,α') \quad \mbox{ for }\quad i=0,1,\ldots, m(G). \]

math.CO↗