A combinatorial proof of Fisher's Inequality

一个划分周长不等式的组合证明

A combinatorial proof of a partition perimeter inequality

本文通过将划分周长与组合相联系，给出了 Amdeberhan-Andrews-Ballantine 定理的组合证明，并表明 Huang 的组合定理可细化 Fu-Tang 的周长定理。

Katsuda-Urakawa不等式的一个扩展

An extension of Katsuda-Urakawa's Faber-Krahn inequality

本文证明了连通有限图的归一化组合p-拉普拉斯算子的第一个狄利克雷特征值最小值仅由tadpole图T_{n,3}取得，扩展了Katsuda-Urakawa的Faber-Krahn不等式，且论证更简洁。

关于某些按奇偶性分开的划分的不等式证明

Combinatorial proof of an inequality on some partitions separated by parity

本文通过组合方法证明了关于按奇偶性分开的划分的不等式在n≥373时成立，回答了Bringmann等人的提问。

An Asymptotic Faber-Krahn Inequality for the Combinatorial Laplacian on Z^2

The Faber-Krahn inequality states that among all open domains with a fixed volume in R^n, the ball minimizes the first Dirichlet eigenvalue of the Laplacian. We study an asymptotic discrete analogue of this for the combinatorial Dirichlet Laplacian acting on induced subgraphs of Z^2. Namely, an induced subgraph G with n vertices is called a minimizing subgraph if it minimizes the first eigenvalue of the combinatorial Dirichlet Laplacian among all induced subgraphs with n vertices. Consider an induced subgraph G and take the interior of the union of closed squares of area 1 about each point of …

Real Analytic Methods in the Formulations of some Combinatorial Inequalities

In this paper, we derive some new combinatorial inequalities by applying well known real analytic results like Hölder's inequality, Young's inequality, and Minkowiski's inequality to the recursively defined sequence $f_n$ of functions \begin{align*} f_0(x) & = χ_{(-1/2, 1/2)} (x), \nonumber f_{n+1}(x) & = f_n(x+1/2)+ f_n(x-1/2), n \in \mathbb{N}\,\cup \,\{0\}. \end{align*} Towards this goal, we derive the closed form of the aforementioned sequence $ (f_n)_{n\in \mathbb{N}\,\cup \,\{0\}}$ of functions and show that it is a sequence of simple functions that are linear combinations of characteris…

A graph inequality on the common neighbourhood

In this note we prove a graph inequality based on the sizes of the common neighbourhoods. We also characterize the extremal graphs that achieve the equality. The result was first discovered as a consequence of the classical Forster's theorem in electric networks. We also present a short combinatorial proof that was inspired by a similar inequality related to the celebrated Turán's theorem.

An extension of Maclaurin's inequalities

We give a combinatorial extension of the classical inequalities of Maclaurin about symmetric functions of several variables. We discuss two problems - one analytical and another combinatorial - and show that they are in some sense equaivalent. These results complete and extend earlier results of Motzkin and Straus, Khadzhiivanov, Sos and Straus, Fisher and Ryan, and Petingi and Rodriguez.

Polynomization of the Liu-Zhang inequality for overpartition function

Let $\overline{p}(n)$ denote the overpartition function. Liu and Zhang showed that $\overline{p}(a) \overline{p}(b)>\overline{p}(a+b)$ for all integers $a,b>1$ by using an analytic result of Engle. We offer in this paper a combinatorial proof to the Liu-Zhang inequaity. More precisely, motivated by the polynomials $P_{n}(x)$ , which generalize the $k$-colored partitions function $p_{-k}(n)$, we introduce the polynomials $\overline{P}_{n}(x)$, which take the number of $k$-colored overpartitions of $n$ as their special values. And by combining combinatorial and analytic approaches, we obtain tha…

A combinatorial proof of the Gaussian product inequality beyond the MTP${}_2$ case

A combinatorial proof of the Gaussian product inequality (GPI) is given under the assumption that each component of a centered Gaussian random vector $\boldsymbol{X} = (X_1, \ldots, X_d)$ of arbitrary length can be written as a linear combination, with coefficients of identical sign, of the components of a standard Gaussian random vector. This condition on $\boldsymbol{X}$ is shown to be strictly weaker than the assumption that the density of the random vector $(|X_1|, \ldots, |X_d|)$ is multivariate totally positive of order $2$, abbreviated MTP${}_2$, for which the GPI is already known to ho…

随机组合优化问题的一个浓度不等式

A concentration inequality for random combinatorial optimisation problems

本文针对随机组合优化问题中的最小权重随机变量，通过定义子集族的可修补性并结合Talagrand不等式与喷洒型论证，证明了其围绕均值的尖锐浓度不等式。

伯杰龙猜想与两个二项式系数的故事

Bergeron's conjecture & a tale of two binomial coefficients

研究伯杰龙猜想，即整数\(1\leq a<b<c<d\)且\(ad=bc\)时高斯多项式系数关系\({\binom{b+c}b}_q \ge {\binom{a+d}a}_q\)。给出两个证明并推广，还给出特殊情况的组合证明。

Grothendieck-type inequalities in combinatorial optimization

We survey connections of the Grothendieck inequality and its variants to combinatorial optimization and computational complexity.

A Detailed Proof of Pohst's Inequality

In 1977 Pohst conjectured a certain inequality for $n$ variables and give a computer-assisted proof for $n\leq 10$. We give a proof for all $n$ using a combinatorial argument. This inequality yields a better bound for the regulator in terms of the discriminant for totally real number fields.

全息熵不等式的组合性质

Combinatorial properties of holographic entropy inequalities

本文提出了一种新的组合框架来研究全息熵不等式，证明了其共享的性质，并推翻了部分先前猜想，提供了新的证据表明所有HEI在时间依赖的全息态中被遵守。

Combinatorial systolic inequalities

We establish combinatorial versions of various classical systolic inequalities. For a smooth triangulation of a closed smooth manifold, the minimal number of edges in a homotopically non-trivial loop contained in the $1$-skeleton gives an integer called the combinatorial systole. The number of top-dimensional simplices in the triangulation gives another integer called the combinatorial volume. We show that a class of smooth manifolds satisfies a systolic inequality for all Riemannian metrics if and only if it satisfies a corresponding combinatorial systolic inequality for all smooth triangulat…

Introduction to the combinatorial atlas

We give elementary self-contained proofs of the strong Mason conjecture recently proved by Anari at. al. (arXiv:1811.01600) and Brändén--Huh (arXiv:1902.03719), and of the classical Alexandrov--Fenchel inequality. Both proofs use the combinatorial atlas technology recently introduced by the authors (arXiv:2110.10740). We also give a formal relationship between combinatorial atlases and Lorentzian polynomials.

通过子模性建立广义信息不等式，以及两个组合问题

Generalized Information Inequalities via Submodularity, and Two Combinatorial Problems

本文通过子模性建立广义信息不等式，推导新的投影不等式，并研究极值图论问题，扩展了已有结果。

Dimension-free L2 maximal inequality for spherical means in the hypercube

We establish the result of the title. In combinatorial terms this has the implication that for sufficiently small eps > 0, for all n, any marking of an eps fraction of the vertices of the n-dimensional hypercube necessarily leaves a vertex x such that marked vertices are a minority of every sphere centered at x.

Systolic inequalities for the number of vertices

Inspired by the classical Riemannian systolic inequality of Gromov we present a combinatorial analogue providing a lower bound on the number of vertices of a simplicial complex in terms of its edge-path systole. Similarly to the Riemannian case, where the inequality holds under a topological assumption of "essentiality", our proofs rely on a combinatorial analogue of that assumption. Under a stronger assumption, expressed in terms of cohomology cup-length, we improve our results quantitatively. We also illustrate our methods in the continuous setting, generalizing and improving quantitatively …