arXiv subjects

Feihu Liu

Publications and source records attributed to Feihu Liu.

At least 19 recordsLinked to original sources

arXiv2026

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

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 simplicial complex. Mu and Welker proved that if the recursive decomposition $f(t)=g(t)+th(t)$ satisfies the corresponding coefficient inequality $h_i<g_i$, then this open problem has an affirmative answer. Mu and Welker also conjectured that the real-rootedness of $f(t)$ implies that of $g(t)$ and $h(t)$. We give counterexamples to the conjecture of Mu and Welker for every degree at least three, and prove that the assertion holds in degrees $1$ and $2$. Moreover, each polynomial we construct is the $f$-polynomial of a simplicial complex.

math.CO↗

arXiv2026

Unimodality for IDP Lattice Simplices of Prime Normalized Volume

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 decomposition property is unimodal. This raises the question of whether the $h^*$-polynomial of a lattice simplex with the integer decomposition property remains unimodal. In this note, we prove that every lattice simplex with the integer decomposition property and prime normalized volume has a unimodal $h^*$-polynomial. Furthermore, we establish several sufficient conditions for the unimodality of the $h^*$-polynomial of such simplices.

math.CO↗

arXiv2026

Real-Rootedness and Gamma-Positivity for a Variation of the Morris Constant Term

Beck and Pixton expressed the Ehrhart polynomial of the Birkhoff polytope as a weighted sum of constant terms of several multivariate rational functions. Xin and Zhang studied a class of constant terms $h_n(t)$, which can be regarded as a variation of the Morris constant term. They proved that $h_n(t)$ is a polynomial of degree $(n-1)^2$ and obtained many nice properties involving the Morris constant term identity. Let $h_n^*(y)=(1-y)^{(n-1)^2+1}\sum_{t\geq0}h_n(t)y^t$. For fixed $n\geq 3$, we obtain the following three main results: (i): $h_n^*(y)$ is a polynomial with positive integer coefficients. (ii): $h_n^*(y)$ is real-rooted. In particular, all its roots are non-positive real numbers. (iii): $h_n^*(y)$ is Gamma-positive. Furthermore, $h_n^*(y)$ is palindromic, unimodal, and ultra log-concave. This confirms Xin and Zhang's conjecture regarding $h_n^*(y)$. As a byproduct, we prove that every root of a Gamma polynomial associated with $h_n^*(y)$ is a negative real number.

math.CO↗

arXiv2026

Proof of a Conjecture on Young Tableaux with Walls

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 work, we prove a conjecture of Fuchs and Yu concerning the enumeration of two classes of three-row Young tableaux with walls. Together with the work of Chang, Fuchs, Liu, Wallner, and Yu, our result verifies a conjecture of Pons and Batle on tree-child networks. This conjecture had been regarded as a specific and challenging problem in the phylogenetics community until its resolution in the present work.

math.CO↗

arXiv2026

Taylor Positivity of Ehrhart Polynomials

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 coefficients $\mathsf{A}_j(P;k)$ in the shifted expansion $L_P(t)=\sum_{j=0}^{d}\mathsf{A}_j(P;k)(t-k)^j$ about a real center $k$. In this paper, we obtain the following four main results. (i) We give exact formulas for these coefficients in terms of the ordinary Ehrhart coefficients, the $h^*$-vector, elementary symmetric functions, and Stirling numbers. (ii) We denote by $τ(P)$ and $τ^+(P)$ the smallest nonnegative integral centers at which all Taylor coefficients are nonnegative and positive, respectively. If $s$ is the degree of the $h^*$-polynomial, then $0\leqτ(P)\leqτ^+(P)\leq\min\{\max\{0,s-1\},\lfloor\frac{d-1}{2}\rfloor\}$. As an application, we slightly improve an upper bound due to Beck, De Loera, Develin, Pfeifle, and Stanley. That is, every real root of $L_P(t)$ lies in $[-d,\lfloor\frac{d-1}{2}\rfloor)$. (iii) Let $ρ(P)$ be the smallest nonnegative real center such that the Taylor coefficients are nonnegative. If $λ_{\mathbb{R}}(f)$ denotes the largest real zero of $f(t)$, with value $-\infty$ when no such zero exists, then $ρ(P)=\max\{0,\max_{0\leq j<d}λ_{\mathbb{R}}\!(L_P^{(j)})\}$. (iv) We establish structural properties of the Taylor coefficients $\mathsf{A}_j(P;k)$, including derivative interlacing, palindromic reflection symmetries, and Laguerre and Newton inequalities. As a final note, these results provide a systematic partial answer to an open problem listed on the website of the American Institute of Mathematics.

math.CO↗

arXiv2026

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

An $(α,β)$ Somos-$4$ sequence $S_n$ is defined by the recurrence $S_nS_{n-4}=αS_{n-1}S_{n-3}+βS_{n-2}^2$ ($n\geq 4$), with suitable initial values, where $α$ and $β$ are constant parameters. A widely studied question is the following: When does the Hankel transform of a generating function become an $(α,β)$ Somos-4 sequence? In particular, how can $α$ and $β$ be derived for such a function? A sufficient condition for this problem has been established by Wang and Zhang. In this paper, we obtain the following three main results. (i): We extend the Wang--Zhang sufficient condition by working over the rational function field. Then we combine this result with the Sulanke--Xin quadratic transformation to resolve all of Barry's currently unsolved $(α,β)$ Somos-4 conjectures, which arise in diverse contexts, including generalized Catalan recurrences, Riordan arrays, generalized Bernstein arrays, and elliptic curves. (ii): We show that the odd and even subsequences of an $(α,β)$ Somos-4 sequence are again $(α,β)$ Somos-4 sequences with transformed parameters. This is employed to establish Barry's Hurwitz transform conjecture. (iii): Using the theory of orthogonal polynomials, we prove a Hankel determinant formula and thereby prove a conjecture related to the $(α,β)$ Somos-4 sequence. In addition, we prove some conjectures on formulas for periodic Hankel determinants.

math.CO↗

arXiv2026

OmniPack: Unified Token Compression for Efficient Omni-modal Large Language Models

Omni-modal large language models (Omni-LLMs) have achieved remarkable performance on audio-visual understanding tasks, but processing long and highly redundant visual and audio token sequences incurs substantial computational overhead, demanding aggressive token compression for efficient deployment. Existing methods often degrade at low token budgets: pre-LLM compression may discard structurally important and globally distributed evidence, whereas inner-LLM compression often underexploits query-conditioned audio-visual collaboration. To address these limitations, we propose OmniPack, a training-free framework that coordinates structural compression before the LLM with task-relevant semantic refinement within the LLM. Before the LLM, OmniPack removes structural redundancy through modality-specific importance, global coverage, and similarity-aware merging. After sufficient multimodal interaction, it further consolidates diverse, task-relevant representations through textual guidance and audio-visual collaboration. Extensive experiments on five benchmarks with three Omni-LLM backbones demonstrate that OmniPack consistently achieves the best performance-efficiency trade-off across diverse retention ratios, outperforming all existing methods. Notably, on Qwen2.5-Omni-7B, OmniPack preserves 98.0% of the original performance while reducing FLOPs to 16.7%, and still retains 92.9% of the original performance with only 6.8% of the original FLOPs.

cs.CV↗

arXiv2026

Counterexamples to the Minimum Period Conjecture for Restricted Partition Functions

For a finite sequence of positive integers $\boldsymbol{a}=(a_1,\dots,a_n)$, the restricted partition function $q_{\boldsymbol{a}}(k)$ denote the number of nonnegative integer solutions to the equation $a_1x_1+a_2x_2+\cdots +a_nx_n=k$. It is proved to be a quasi-polynomial of degree $n-1$. Write $q_{\boldsymbol{a}}(k)=\sum_{j=0}^{n-1}c_j(k)k^j$ with periodic coefficient functions $c_j$, and set $b_m=\#\{i:m\mid a_i\}$. In 2008, Beck, Sam, and Woods conjectured that the minimum period of $c_j(k)$ is $\mathrm{lcm}\{m:b_m>j\}$. In this paper, we derive an exact root-of-unity formula for every coefficient function $c_j(k)$. The formula proves the conjectured divisibility upper bound, but it also reveals a lower bound for the period of $c_j(k)$. Both divisibility bounds are sharp. This leads us to construct a family of counterexamples to this conjecture.

math.CO↗

arXiv2026

On the Frobenius Number of Quotients of Numerical Semigroups

Given a numerical semigroup $S$ and a positive integer $p$, the quotient $\frac{S}{p}=\{n\in \mathbb{N} \mid pn\in S\}$ also forms a numerical semigroup. When $S=\langle a,b\rangle$ with $\gcd(a,b)=1$, a well-known open problem is to find a closed-form formula for the Frobenius number $g\!\left(\frac{\langle a,b\rangle}{p}\right)$, which remains open even in the special case $b=a+1$. Inspired by Curtis's theorem on the non-existence of polynomial formulas for the Frobenius number $g(\langle s_1,s_2,s_3\rangle)$, we provide a negative answer to this open problem in a certain sense. Concretely, we obtain the following three main results. (i): The Frobenius number $g\!\left(\frac{\langle a,b\rangle}{p}\right)$ cannot be represented, uniformly in $a,b,p$, by any finite collection of polynomial (or rational) formulas. (ii): For each fixed $p$, the function $a\mapsto g\!\left(\frac{\langle a,a+1\rangle}{p}\right)$ is a quadratic quasi-polynomial with period dividing $p$. (iii): There is no nonzero polynomial $F\in \mathbb{C}[X_1,X_2,X_3]$ satisfying $F\left(a,p,g\!\left(\frac{\langle a,a+1\rangle}{p}\right)\right)=0$ for all primes $a,p$ with $2<p<a$; the same conclusion already holds if only $p$ is required to be prime and $a$ ranges over all integers greater than $p$. While (iii) is stronger than (i), the proofs of the two results reveal different insights. Dirichlet's theorem on primes in arithmetic progressions plays a crucial role in our arguments.

math.NT↗

arXiv2026

Proof of Barry's Four Hankel Determinant Conjectures

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 these four conjectures. The proofs are unified within a common algebraic framework: we interpret the Hankel determinants as Gram determinants and use a basis of shifted monic Chebyshev polynomials to reveal the finite-band structure of the associated Gram matrices.

math.CO↗

arXiv2026

An $O(n\log^2n)$ Algorithm for Computing Hankel Determinants up to Order $n$

Given the rational power series $h(x) = \sum_{i \geq 0} h_i x^i \in \mathbb{C}[[x]]$, the Hankel determinant of order $n$ is defined as $H_n(h(x)) := \det (h_{i+j})_{1 \leq i,j \leq n}$. We explore the relationship between the Hankel continued fraction and the generalized Sturm sequence. This connection inspires the development of a novel algorithm for computing the Hankel determinants $\{H_i(h(x))\}_{i=0}^{n-1}$ using $O(n \log^2 n)$ arithmetic operations. We also explore the connection between the generalized Sturm sequences and the signature of Hankel matrices.

math.CO↗

arXiv2026

Magic Positivity for the Ehrhart Polynomials of Partial Permutohedra

For positive integers $m,n$, the partial permutohedron $\mathcal{P}(m,n)$ is a lattice polytope constructed as the convex hull of vectors in $\{0, 1, \dots, n\}^m$ that have distinct non-zero entries. We prove that for $n \ge m-1$, the Ehrhart polynomial of $\mathcal{P}(m,n)$ is magic positive except for the single case $(m,n)=(2,1)$. In particular, the Ehrhart polynomial of the parking function polytope (integrally equivalent to $\mathcal{P}(m,m-1)$) is magic positive for $m \ge 3$. For $n<m-1$, we discuss the magic positivity of the Ehrhart polynomial of $\mathcal{P}(m,n)$ for $n=1,2,3$. There exist infinitely many counterexamples with $n<m-1$ showing that the Ehrhart polynomial of $\mathcal{P}(m,n)$ is not magic positive. This partially resolves an open problem proposed by Ferroni and Higashitani.

math.CO↗

arXiv2026

Ehrhart Theory of the Join of Two Lattice Polytopes

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 open problem listed on the website of the American Institute of Mathematics. This paper resolves this open problem. We first construct counterexamples showing that the join of two Ehrhart positive polytopes is not necessarily Ehrhart positive. Then we prove that if two lattice polytopes have the integer decomposition property and the spanning property, then their join also has these two properties. However, the very ample property is not inherited under joins. Finally, we show that unimodular triangulations, regular triangulations, and quadratic triangulations are preserved under the join operation. As a byproduct, we state the necessary and sufficient condition for the Cartesian product of two Gorenstein lattice polytopes to remain Gorenstein.

math.CO↗

arXiv2026

Explicit Construction of Polytopes whose Ehrhart Polynomials Realize any Given Sign Pattern

In Ehrhart theory, the well-known sign pattern problem asks: given a positive integer $d\geq 3$ and integers $1 \leq i_1 < \cdots < i_k \leq d-2$, does there exist a $d$-dimensional integral polytope $\mathcal{P}$ such that in its Ehrhart polynomial $i(\mathcal{P}, t)$ the coefficients of $t^{i_1}, \ldots, t^{i_k}$ are negative, while all remaining coefficients are positive? This problem was proposed by Hibi, Higashitani, Tsuchiya, and Yoshida. In this paper, we first construct a class of simplices $\mathcal{S}_d(m)$ whose Ehrhart polynomial has leading coefficient $m$ and all other coefficients fixed positive constants. Then, using the Cartesian product of $\mathcal{S}_d(m)$ and the Reeve tetrahedron, we obtain the first complete solution to the sign pattern problem. Finally, while attacking the sign pattern problem, we discovered a fast algorithm for computing the $h^*$-polynomial of a class of simplices $Δ(0,q)$. This algorithm is crucial for constructing the simplices $\mathcal{S}_d(m)$.

math.CO↗

arXiv2026

Proofs of four generating function conjectures for arbor polytopes

This paper proves four conjectured generating series, due to Chapoton, which concern invariants of posets and polytopes associated with a specific sequence of arbors. Two of these conjectures provide closed-form formulas for the generating series of the Zeta polynomial and the generating series of the M-triangle of the poset, respectively. The remaining two conjectures pertain, respectively, to the Ehrhart polynomial and the Laplace transform of the volume function of the associated arbor polytope.

math.CO↗

arXiv2026

The Frobenius problem for a class of quotients of numerical semigroups

Given a numerical semigroup $S$ and a positive integer $p$, the quotient $\frac{S}{p}=\{x\in \mathbb{N} \mid px\in S\}$ also forms a numerical semigroup. In this paper, we first characterize the Apéry set for a class of quotients of numerical semigroups. Under certain conditions, we then derive half-closed form formulas for their Frobenius number and genus. Furthermore, for specific values of part parameters, we obtain explicit formulas for the Frobenius number of certain quotients of numerical semigroups.

math.CO↗

arXiv2026

Unimodular Equivalence of Integral Simplices

Testing the unimodular equivalence of two full-dimensional integral simplices can be reduced to testing unimodular permutation (UP) equivalence of two nonsingular matrices. We conduct a systematic study of UP-equivalence, which leads to the first average-case quasi-polynomial time algorithm, called \texttt{HEM}, for deciding the unimodular equivalence of $d$-dimensional integral simplices, as well as achieving a polynomial-time complexity with a failure probability less than $2.5 \times 10^{-7}$. A key ingredient is the introduction of the \emph{permuted Hermite normal form} and its associated \emph{pattern group}, which streamlines the UP-equivalence test by comparing canonical forms derived from induced coset representatives. We also present an acceleration strategy based on Smith normal forms. As a theoretical by-product, we prove that two full-dimensional integral simplices are unimodularly equivalent if and only if their $n$-dimensional pyramids are unimodularly equivalent. This resolves an open question posed by Abney-McPeek et al.

math.CO↗

arXiv2025

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

MacMahon introduced partition analysis in his book ``Combinatory Analysis'' as a computational technique for solving problems related to systems of linear Diophantine equations and inequalities. This paper aims to develop a fundamental computational method for MacMahon's partition analysis. As applications, we present simplified computations for ``Han's formula'', the ``$k$-gon partitions problem'', and the ``two-dimensional problem''. Moreover, we apply our method to solve a challenging problem.

math.CO↗