arXiv subjects

Suijie Wang

Publications and source records attributed to Suijie Wang.

32 records · Page 2Linked to original sources

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 \[ a_i(G,α)\le a_i(G,α') \quad \mbox{ for }\quad i=0,1,\ldots, m(G). \]

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arXiv2024

Two Bijections on NBC Subsets

We establish two explicit bijections: from acyclic reorientations of an oriented matroid to no broken circuit (NBC) subsets of its underlying matroid, and from regions of a real hyperplane arrangement to its affine NBC subsets.

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arXiv2024

Deformed Intersections of Half-spaces

This paper is devoted to the classification problems concerning extended deformations of convex polyhedra and real hyperplane arrangements in the following senses: combinatorial equivalence of face posets, normal equivalence on normal fans of convex polyhedra, and sign equivalence on half-spaces. The extended deformations of convex polyhedra arise from parallel translations of given half-spaces and hyperplanes, whose normal vectors give rise to the so-called ``derived arrangement'' proposed by Rota as well as Crapo in different forms. We show that two extended deformations of convex polyhedra are normally (combinatorially, as a consequence) equivalent if they are parameterized by the same open face of the derived arrangement. Note that these extended deformations are based on parallel translations of the given hyperplanes. It allows us to study three deformations of real hyperplane arrangements: parallel translations, conings, and elementary lifts, whose configuration spaces are parameterized by open faces of the derived arrangement. Consequently, it gives a characterization of the normal, combinatorial, and sign equivalences of those three deformations via the derived arrangement. Additionally, the relationships among these three equivalence relations are discussed, and several new descriptions of real derived arrangements associated with faces and sign vectors of real hyperplane arrangements are provided.

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arXiv2023

One-element Extensions of Hyperplane Arrangements

We classify one-element extensions of a hyperplane arrangement by the induced adjoint arrangement. Based on the classification, several kinds of combinatorial invariants including Whitney polynomials, characteristic polynomials, Whitney numbers and face numbers, are constants on those strata associated with the induced adjoint arrangement, and also order-preserving with respect to the intersection lattice of the induced adjoint arrangement. As a byproduct, we obtain a convolution formula on the characteristic polynomials $χ(\mathcal{A}+H_{\bmα,a},t)$ when $\mathcal{A}$ is defined over a finite field $\mathbb{F}_q$ or a rational arrangement.

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arXiv2023

Adjoints of Matroids

We show that an adjoint of a loopless matroid is connected if and only if it itself is connected. Our first goal is to study the adjoint of modular matroids. We prove that a modular matroid has only one adjoint (up to isomorphism) which can be given by its opposite lattice, and proceed to present some alternative characterizations of modular matroids associated to adjoints and opposite lattices. The other purpose is to investigate the adjoint sequence $ad^0M,adM,ad^2M,\ldots$ of a connected matroid $M$. We classify such adjoint sequences into three types: finite, cyclic and convergent. For the first two types, the adjoint sequences eventually stabilize at the finite projective geometries except for free matroids. For the last type, the infinite non-repeating adjoint sequences are convergent to the infinite projective geometries.

math.CO↗

arXiv2020

Parallel Translates of Represented Matroids

Given an $\Bbb{F}$-represented matroid $(M,ρ)$ with the ground set $[m]$, the representation $ρ$ naturally defines a hyperplane arrangement $\mathcal{A}_ρ$. We will study its parallel translates $\mathcal{A}_{ρ,{g}}$ of $\mathcal{A}_ρ$ for all ${ g}\in \mathbb{F}^m$. Its intersection semi-lattices $L(\mathcal{A}_{ρ,{ g}})$ and the characteristic polynomials $χ(\mathcal{A}_{ρ,{ g}},t)$ will be classified by the intersection lattice of the derived arrangement $\mathcal{A}_{δρ}$, which is a hyperplane arrangement associated with the derived matroid $(δM,δρ)$ and also known as the discriminantal arrangement in the literature. As a byproduct, we obtain a comparison result and a decomposition formula on the characteristic polynomials $χ(\mathcal{A}_{ρ,{ g}},t)$.

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arXiv2020

Bijections on $r$-Shi and $r$-Catalan Arrangements

Associated with the $r$-Shi arrangement and $r$-Catalan arrangement in $\Bbb{R}^n$, we introduce a cubic matrix for each region to establish two bijections in a uniform way. Firstly, the positions of minimal positive entries in column slices of the cubic matrix will give a bijection from regions of the $r$-Shi arrangement to $O$-rooted labeled $r$-trees. Secondly, the numbers of positive entries in column slices of the cubic matrix will give a bijection from regions of the $r$-Catalan arrangement to pairings of permutation and $r$-Dyck path. Moreover, the numbers of positive entries in row slices of the cubic matrix will recover the Pak-Stanley labeling, a celebrated bijection from regions of the $r$-Shi arrangement to $r$-parking functions.

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arXiv2018

Pattern Avoidance of Generalized Permutations

In this paper, we study pattern avoidances of generalized permutations and show that the number of all generalized permutations avoiding $π$ is independent of the choice of $π\in S_3$, which extends the classic results on permutations avoiding $π\in S_3$. Extending both Dyck path and Riordan path, we introduce the Catalan-Riordan path which turns out to be a combinatorial interpretation of the difference array of Catalan numbers. As applications, we interpret both Motzkin and Riordan numbers in two ways, via semistandard Young tableaux of two rows and generalized permutations avoiding $π\in S_3$. Analogous to Lewis's method, we establish a bijection from generalized permutations to rectangular semistandard Young tableaux which will recover several known results in the literature.

math.CO↗

arXiv2018

Order Preserving Maps of Posets

For any two finite posets $P$ and $Q$, let $\Hom(P,Q)$ be the hom-poset consisting of all order preserving maps from $P$ to $Q$, and $J(Q)$ the collection of all order ideals of $Q$. In this paper, we study some basic properties of the hom-poset $\Hom(P,Q)$ and prove that $\Hom\big(P,J(Q)\big)$ is a distributive lattice and characterized by \[ \Hom\big(P,J(Q)\big)\cong J(P^*\times Q), \] where $P^*$ is the dual of $P$. Consequently, we obtain that $\Hom\big(P,J(Q)\big)$ and $\Hom\big(Q,J(P)\big)$ are dual isomorphic, i.e., \[ \Hom\big(P,J(Q)\big)\cong \Hom^{*}\big(Q,J(P)\big). \] As applications, we calculate the number of order preserving maps from any poset to the boolean algebra, and the characteristic polynomial of $\Hom\big(P,J(Q)\big)$.

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arXiv2017

Orbits of antichains in certain root posets

This paper gives another proof of Propp and Roby's theorem saying that the average antichain size in any reverse operator orbit of the poset $[m]\times [n]$ is $\frac{mn}{m+n}$. It is conceivable that our method should work for other situations. As a demonstration, we show that the average size of antichains in any reverse operator orbit of $[m]\times K_{n-1}$ equals $\frac{2mn}{m+2n-1}$. Here $K_{n-1}$ is the minuscule poset $[n-1]\oplus ([1] \sqcup [1]) \oplus [n-1]$. Note that $[m]\times [n]$ and $[m]\times K_{n-1}$ can be interpreted as sub-families of certain root posets. We guess these root posets should provide a unified setting to exhibit the homomesy phenomenon defined by Propp and Roby.

math.CO↗

arXiv2015

Bounds on Characteristic Polynomials

Suppose $G$ is a simple graph with $n$ vertices, $m$ edges, and rank $r$. Let $χ_G(t)=a_0t^n-a_1t^{n-1}+\cdots +(-1)^ra_rt^{n-r}$ be the chromatic polynomial of $G$. For $q,k\in \Bbb{Z}$ and $0\le k\le q+r+1$, we obtain a sharp two-side bound for the partial binomial sum of the coefficient sequence, that is, \[ {r+q\choose k}\le \sum_{i=0}^{k}{q\choose k-i}a_{i}\le {m+q\choose k}. \] Indeed, this bound holds for the characteristic polynomial of hyperplane arrangements and matroids, and its weak version can be generalized to the characteristic polynomial of toric arrangements and arithmetic matroids. We also propose a problem on the geometric interpretation of the above bound.

math.CO↗

arXiv2012

Mobius Conjugation and Convolution Formulae

Let $P$ be a locally finite poset with the interval space $\Int(P)$, and $R$ a ring with identity. We shall introduce the Möbius conjugation $μ^\ast$ sending each function $f:P\to R$ to an incidence function $μ^\ast(f):\Int(P)\to R$ such that $μ^\ast(fg)=μ^\ast(f)\astμ^\ast(g)$. Taking $P$ to be the intersection poset of a hyperplane arrangement $\mathcal{A}$, we shall obtain a convolution identity for the number $r(\mathcal{A})$ of regions and the number $b(\mathcal{A})$ of relatively bounded regions, and a reciprocity theorem of the characteristic polynomial $χ(\mathcal{A},t)$, which also leads to a combinatorial interpretation to the values $|χ(\mathcal{A},-q)|$ for large primes $q$. Moreover, all known convolution identities on Tutte polynomials of matroids will be direct consequences after specializing the poset $P$ and functions $f,g$.

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arXiv2011

Laplacian spectral characterization of some graph products

This paper studies the Laplacian spectral characterization of some graph products. We consider a class of connected graphs: $\mathscr{G}={G : |EG|\leq|VG|+1}$, and characterize all graphs $G\in\mathscr{G}$ such that the products $G\times K_m$ are $L$-DS graphs. The main result of this paper states that, if $G\in\mathscr{G}$, except for $C_{6}$ and $Θ_{3,2,5}$, is $L$-DS graph, so is the product $G\times K_{m}$. In addition, the $L$-cospectral graphs with $C_{6}\times K_{m}$ and $Θ_{3,2,5}\times K_{m}$ have been found.

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