Feihu Liu

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.HO2026

Ehrhart Properties under Operations on Lattice Polytopes

Feihu Liu

This paper investigates the preservation of three classes of Ehrhart properties under various operations on lattice polytopes. These operations include Cartesian products, lattice…

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,…

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…

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…

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…

Hankel Transform and $(α, β)$ Somos-4 Sequences

Feihu Liu, Ying Wang, Zihao Zhang

An $(α, β)$ Somos- $4$ sequence $S_{n}$ is defined by the recurrence $S_{n} S_{n - 4} = α S_{n - 1} S_{n - 3} + β S_{n - 2}^{2}$ ( $n \geq 4$ ), with suitable initial values, where $α$ and $β$ are constant p…

Counterexamples to the Minimum Period Conjecture for Restricted Partition Functions

Feihu Liu, Jinlong Tang, Guoce Xin +1

For a finite sequence of positive integers $a = (a_{1}, \dots, a_{n})$ , the restricted partition function $q_{a} (k)$ denote the number of nonnegative integer solu…

Proof of Barry's Four Hankel Determinant Conjectures

Feihu Liu, Ying Wang, Zihao Zhang

Barry introduced a central transform of integer sequences and proposed four conjectures concerning the Hankel transforms of central transform of four rational families. We prove th…

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…

Ehrhart Theory of the Join of Two Lattice Polytopes

Feihu Liu, Sihao Tao, Guoce Xin

Inspired by research on the Cartesian product of two lattice polytopes, this paper investigates the Ehrhart theory of the join of two lattice polytopes. This is also a well-known o…

Unimodular Equivalence of Integral Simplices

Feihu Liu, Sihao Tao, Guoce Xin

Testing the unimodular equivalence of two full-dimensional integral simplices can be reduced to testing unimodular permutation (UP) equivalence of two nonsingular matrices. We cond…

Proof of a Conjecture on Young Tableaux with Walls

Zhicong Lin, Feihu Liu, Jiahang Liu +2

Banderier, Marchal, and Wallner considered Young tableaux with walls, which are similar to standard Young tableaux, except that local decreases are allowed at some walls. In this w…

MacMahon's $Ω_{\geq}$ operator: A computational framework

Feihu Liu, Guoce Xin

MacMahon introduced partition analysis in his book ``Combinatory Analysis'' as a computational technique for solving problems related to systems of linear Diophantine equations and…

The Sign Pattern Problem for Ehrhart Polynomials

Feihu Liu, Sihao Tao, Guoce Xin

We investigate the sign patterns of coefficients in the Ehrhart polynomial of the Cartesian product between the $r$ -th pyramid over the Reeve tetrahedron and the hypercube $[0, n]^…

Enumeration of Corona for Lozenge Tilings

Craig Knecht, Feihu Liu, Guoce Xin

Knecht considers the enumeration of coronas. This is a counting problem for two specific types of lozenge tilings. Their exact closed formulas are conjectured in [A380346] and [A38…

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…

Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS

Feihu Liu, Guoce Xin, Chen Zhang

In this paper, we provide an overview of Ehrhart polynomials associated with order polytopes of finite posets, a concept first introduced by Stanley. We focus on their combinatoria…

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…

On $P$ -partitions Extended by Two-Rowed Plane Partitions

Jingxuan Li, Feihu Liu, Guoce Xin

Inspired by Gansner's elegant $k$ -trace generating function for rectangular plane partitions, we introduce two novel operators, $φ_{z}$ and $ψ_{z}$ , along with their combinatorial…

Proof of a conjecture on graph polytope

Feihu Liu

Graph polytopes arising from vertex-weighted graphs were first introduced by Bóna, Ju, and Yoshida. We prove a conjecture stating that for any simple connected graph, the numerator…

A Fast Algorithm for Denumerants with Three Variables

Feihu Liu, Guoce Xin

Let $a, b, c$ be distinct positive integers such that $a < b < c$ and $g cd (a, b, c) = 1$ . For any non-negative integer $n$ , the denumerant function $d (n; a, b, c)$ denotes the number of soluti…

math.CO2024★ 3 cited

Three Simple Reduction Formulas for the Denumerant Functions

Feihu Liu, Guoce Xin, Chen Zhang

Let $A$ be a nonempty set of positive integers. The restricted partition function $p_{A} (n)$ denotes the number of partitions of $n$ with parts in $A$ . When the elements in $A$ are p…