Zi-Hao Zhang

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20212026

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20 papers · 1 filter

math.CO2026

Ehrhart positivity for paving positroids

Feihu Liu, Zihao Zhang

A central conjecture of Ferroni, Jochemko, and Schröter posits that every positroid is Ehrhart positive. While Ehrhart positivity has recently been verified for notable classes suc…

math.CO2026

Two families of Hermite normal form simplices

Feihu Liu, Jinlong Tang, Sihao Tao +1

We study Ehrhart coefficients, the integer decomposition property, and unimodular triangulations for two families of Hermite normal form simplices considered by Bruckamp, Caicedo,…

math.CO2026

Counterexamples to the Mu--Welker recursive decomposition in every degree

Feihu Liu, Ying Wang, Zihao Zhang

The well-known open problem of Bell and Skandera asks whether a real-rooted polynomial $f (t)$ with positive integer coefficients and constant term one is the $f$ -polynomial of a si…

math.CO2026

Unimodality for IDP Lattice Simplices of Prime Normalized Volume

Feihu Liu, Zihao Zhang

Recently, Ferroni constructed a family of counterexamples to the well-known conjecture in Ehrhart theory stating that the $h^{*}$ -polynomial of a lattice polytope with the integer de…

math.CO2026

Taylor Positivity of Ehrhart Polynomials

Feihu Liu, Zihao Zhang

Let $P$ be a $d$ -dimensional lattice polytope with Ehrhart polynomial $L_{P} (t)$ . Motivated by the study of Ehrhart positivity and magic positivity, we investigate the Taylor coeffic…

math.CO2026

The Ehrhart series of magic squares of orders seven and eight

Dun Qiu, Guoce Xin, Zihao Zhang

Let $IMS_{n} (m)$ count the $n \times n$ nonnegative integer matrices whose row sums, column sums, main-diagonal sum, and antidiagonal sum are all $m$ . We determine the Ehrhar…

math.CO2026

A polynomial time algorithm for almost bounded denumerant

Guoce Xin, Chen Zhang, Zihao Zhang

Sylvester's denumerant $d (t; A)$ counts the number of nonnegative integer solutions to $\sum_{i = 1}^{N} a_{i} x_{i} = t$ , where $A = (a_{1}, \dots, a_{N})$ is a se…

math.CO2026

Polynomial-Time Lattice-Point Counting without Barvinok Decomposition

Guoce Xin, Zihao Zhang

By using constant term manipulations, we present the first polynomial-time algorithm for lattice-point counting in fixed dimension that does not rely on Barvinok's unimodular decom…

math.CO2026

Polynomial-Time Evaluation of Aardal-Lenstra Denumerants via Constant Term Method

Jinlong Tang, Guoce Xin, Zihao Zhang

Aardal and Lenstra systematically studied hard knapsack problems of the form $a_{1} x_{1} + \dots + a_{n} x_{n} = b$ , where $a_{i} = p_{i} M + r_{i} N$ , $(M, N)$ is a coprime pair of positive integers, and th…

math.CO2026

Magic Positivity for the Ehrhart Polynomials of Partial Permutohedra

Feihu Liu, Zihao Zhang

For positive integers $m,n$, the partial permutohedron $P (m, n)$ is a lattice polytope constructed as the convex hull of vectors in ${0, 1, \dots, n}^{m}$ that have dist…

math.CO2025

On the quantum chromatic number of Hamming and generalized Hadamard graphs

Xiwang Cao, Keqin Feng, Hexiang Huang +2

As a fundamental metric for quantifying quantum advantage in non-local games, the quantum chromatic number reveals the power of entanglement in distributed tasks. In this paper, we…

math.CO2025

Closed-Form Decomposition for Simplicial Cones and PDBarv Algorithm for Lattice Point Counting

Sihao Tao, Guoce Xin, Zihao Zhang

Counting lattice points within a rational polytope is a foundational problem with applications across mathematics and computer science. A key approach is Barvinok's algorithm, whic…

math.CO2025

A combinatorial simplicial cone decomposition

Guoce Xin, Xinyu Xu, Zihao Zhang

This paper introduces an algebraic combinatorial approach to simplicial cone decompositions, a key step in solving inhomogeneous linear Diophantine systems and counting lattice poi…

math.CO2025

An $O (n lo g^{2} n)$ Algorithm for Computing Hankel Determinants up to Order $n$

Feihu Liu, Guoce Xin, Zihao Zhang

Given the rational power series $h (x) = \sum_{i \geq 0} h_{i} x^{i} \in C [[x]]$ , the Hankel determinant of order $n$ is defined as $H_n(h(x)) := \det (h_{i+j})_{1 \leq i,j \le…

math.CO2024

Order Polytopes of Dimension $\leq 13$ are Ehrhart Positive

Feihu Liu, Guoce Xin, Zihao Zhang

The order polytopes arising from the finite poset were first introduced and studied by Stanley. For any positive integer $d \geq 14$ , Liu and Tsuchiya proved that there exists a non…

math.CO2024

A Combinatorial Decomposition of Knapsack Cones

Guoce Xin, Yingrui Zhang, Zihao Zhang

In this paper, we focus on knapsack cones, a specific type of simplicial cones that arise naturally in the context of the knapsack problem $x_{1} a_{1} + \dots + x_{n} a_{n} = a_{0}$ . We pr…

math.CO2023

Algebraic Volume for Polytope Arise from Ehrhart Theory

Guoce Xin, Xinyu Xu, Yingrui Zhang +1

Volume computation for $d$ -polytopes $P$ is fundamental in mathematics. There are known volume computation algorithms, mostly based on triangulation or signed-decompositi…

math.CO2023

Fast Evaluation of Generalized Todd Polynomials: Applications to MacMahon's Partition Analysis and Integer Programming

Guoce Xin, Yingrui Zhang, ZiHao Zhang

The Todd polynomials, denoted as $t d_{k} (b_{1}, b_{2}, \dots, b_{m})$ , are characterised by their generating functions: $k \geq 0 \sum t d_{k} s^{k} = i = 1 \prod m \frac{b _{i} s}{e ^{b_{i} s} - 1} .$ …

math.CO2021

Proof of a Conjecture on Hankel Determinants for Dyck Paths with Restricted Peak Heights

Guoce Xin, Zihao Zhang

For any integer $m \geq 2$ and $r \in {1, \dots, m}$ , let $f_{n}^{m, r}$ denote the number of $n$ -Dyck paths whose peak's heights are $im + r$ for some integer $i$ . We find the generati…

math.CO2021

Enumerating Magic Distinct Labellings of the Cube

Guoce Xin, Yingrui Zhang, Zihao Zhang

We find by applying MacMahon's partition analysis that all magic labellings of the cube are of eight types, each generated by six basis elements. A combinatorial proof of this fact…

Ehrhart positivity for paving positroids

Two families of Hermite normal form simplices

Counterexamples to the Mu--Welker recursive decomposition in every degree

Unimodality for IDP Lattice Simplices of Prime Normalized Volume

Taylor Positivity of Ehrhart Polynomials

The Ehrhart series of magic squares of orders seven and eight

A polynomial time algorithm for almost bounded denumerant

Polynomial-Time Lattice-Point Counting without Barvinok Decomposition

Polynomial-Time Evaluation of Aardal-Lenstra Denumerants via Constant Term Method

Magic Positivity for the Ehrhart Polynomials of Partial Permutohedra

On the quantum chromatic number of Hamming and generalized Hadamard graphs

Closed-Form Decomposition for Simplicial Cones and PDBarv Algorithm for Lattice Point Counting

A combinatorial simplicial cone decomposition

An O(nlog2n) Algorithm for Computing Hankel Determinants up to Order n

Order Polytopes of Dimension ≤13 are Ehrhart Positive

A Combinatorial Decomposition of Knapsack Cones

Algebraic Volume for Polytope Arise from Ehrhart Theory

Fast Evaluation of Generalized Todd Polynomials: Applications to MacMahon's Partition Analysis and Integer Programming

Proof of a Conjecture on Hankel Determinants for Dyck Paths with Restricted Peak Heights

Enumerating Magic Distinct Labellings of the Cube

An $O (n lo g^{2} n)$ Algorithm for Computing Hankel Determinants up to Order $n$

Order Polytopes of Dimension $\leq 13$ are Ehrhart Positive