Igor Pak
This is an expanded version of the Notices of the AMS column with the same title. The text is unchanged, but we added acknowledgements and a large number of endnotes which provide the context and the references.
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We use a probabilistic method to produce some combinatorial inequalities by considering pattern containment in permutations and words.
A Unified Treatment of Some Classic Combinatorial Inequalities Using the Variance Method
本文介绍方差方法在设计理论和编码理论经典不等式证明中的应用，在统一框架下回顾展示相关不等式，并结合作者自身研究实例加以说明。
Let M and N be Orlicz functions. We establish some combinatorial inequalities and show that the product spaces l^n_M(l^n_N) are uniformly isomorphic to subspaces of L_1 if M and N are "separated" by a function t^r, 1<r<2.
We survey connections of the Grothendieck inequality and its variants to combinatorial optimization and computational complexity.
In 1981, Stanley applied the Aleksandrov-Fenchel inequalities to prove a logarithmic concavity theorem for regular matroids. Using ideas from electrical network theory we prove a generalization of this for the wider class of matroids with the ``half-plane property''. Then we explore a nest of inequalities for weighted basis-generating polynomials that are related to these ideas. As a first result from this investigation we find that every matroid of rank three or corank three satisfies a condition only slightly weaker than the conclusion of Stanley's theorem.
Faber-Krahn inequalities for first Dirichlet eigenvalues of combinatorial $p$-Laplacian on graphs with boundary
本文研究了图上组合p-拉普拉斯算子的第一狄利克雷特征值的费伯-克兰不等式，证明当p>1时，具有n个顶点或n条边的连通图的最小第一p-狄利克雷特征值仅在 tadpole 图T_{n,3}上取得。
Using calculus we show how to prove some combinatorial inequalities of the type log-concavity or log-convexity. It is shown by this method that binomial coefficients and Stirling numbers of the first and second kinds are log-concave, and that Motzkin numbers and secondary structure numbers of rank 1 are log-convex. In fact, we prove via calculus a much stronger result that a natural continuous ``patchwork'' (i.e. corresponding dynamical systems) of Motzkin numbers and secondary structures recursions are increasing functions. We indicate how to prove asymptotically the log-convexity for general…
We consider the number of the $6$-regular partitions of $n$, $b_6(n)$, and give infinite families of congruences modulo $3$ (in arithmetic progression) for $b_6(n)$. We also consider the number of the partitions of $n$ into distinct parts not congruent to $\pm 2$ modulo $6$, $Q_2(n)$, and investigate connections between $b_6(n)$ and $Q_2(n)$ providing new combinatorial interpretations for these partition functions. In this context, we discover new infinite families of linear inequalities involving Euler's partition function $p(n)$. Infinite families of linear inequalities involving the $6$-reg…
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…
On Two Combinatorial Inequalities That Explain the Blimpy Shape of Heady-s and Taily-s Bit Strings
研究关于特定评分系统下的两个组合不等式，用生成函数证明，揭示了位串图形零分附近单峰性及正负分形状不对称性，还给出离散随机变量期望值下界等结论。
Small ball inequalities have been extensively studied in the setting of Gaussian processes and associated Banach or Hilbert spaces. In this paper, we focus on studying small ball probabilities for sums or differences of independent, identically distributed random elements taking values in very general sets. Depending on the setting--abelian or nonabelian groups, or vector spaces, or Banach spaces--we provide a collection of inequalities relating different small ball probabilities that are sharp in many cases of interest. We prove these distribution-free probabilistic inequalities by showing th…
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.
Combinatorial anti-concentration inequalities, with applications
本文证明了独立伯努利随机变量函数的多种反集中不等式，包括泊松型定理和非负系数多项式的不等式，推广了经典定理，并应用于随机图子图计数。
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…
This paper is about hyperbolic properties on planar graphs. First, we study the relations among various kinds of strong isoperimetric inequalities on planar graphs and their duals. In particular, we show that a planar graph satisfies a strong isoperimetric inequality if and only if its dual has the same property, if the graph satisfies some minor regularity conditions and we choose an appropriate notion of strong isoperimetric inequalities. Second, we consider planar graphs where negative combinatorial curvatures dominate, and use the outcomes of the first part to strengthen the results of Hig…
Several combinatorial inequalities related to squarefree monomial ideals
本文利用Hilbert深度作为无平方单项式理想商的Stanley深度上界这一事实，证明了几个涉及多项式系数的组合不等式。
How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically non-trivial closed curves, pants decompositions, and cut graphs with a given combinatorial map in triangulated combinatorial surfaces (or their dual cross-metric counterpart). Our work builds upon Riemannian systolic inequalities, which bound the minimum length of non-trivial closed curves in terms of the genus and the area of the surface. We first describe a systematic way to translate Riemannian systolic inequalities to a discrete…
Combinatorial proofs of inequalities involving the number of partitions with parts separated by parity
本文组合证明了具有按奇偶性分离部分的分拆数之间的几个不等式，并证明了 Fu 和 Tang 提出的一个关于部分重数受限分拆的猜想不等式。
We consider log-convex sequences that satisfy an additional constraint imposed on their rate of growth. We call such sequences log-balanced. It is shown that all such sequences satisfy a pair of double inequalities. Sufficient conditions for log-balancedness are given for the case when the sequence satisfies a two- (or more-) term linear recurrence. It is shown that many combinatorially interesting sequences belong to this class, and, as a consequence, that the above-mentioned double inequalities are valid for all of them.
In a recent work [2] with Datta, we introduced the mu vector (with respect to a given field) of simplicial complexes and used it to study tightness and lower bounds. In this paper, we modify the definition of mu vectors. With the new definition, most results of [2] become correct without the hypothesis of 2-neighbourliness. In particular, the combinatorial Morse inequalities of [2] are now true of all simplicial complexes. As an application, we prove the following generalized lower bound theorem (GLBT) for connected locally tame combinatorial manifolds. If $M$ is such a manifold of dimension $…
Concentration results and probabilistic analysis for combinatorial problems like the TSP, MWST, graph coloring have received much attention, but generally, for i.i.d. samples (i.i.d. points in the unit square for the TSP, for example). Here, we prove two probability inequalities which generalize and strengthen Martingale inequalities. The inequalities provide the tools to deal with more general heavy-tailed and inhomogeneous distributions for combinatorial problems. We prove a wide range of applications - in addition to the TSP, MWST, graph coloring, we also prove more general results than kno…
Log-concave poset inequalities
本文研究 matroids、polymatroids 等集合系统的组合不等式，利用纯 linear algebra 方法证明并扩展了加权可行字的 log-concavity 不等式及等式条件，并推广了 Stanley 不等式。
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。
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。
In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a complex, and the spectrum of the high dimensional Laplacian defined by Eckmann. In particular, we present a Cheeger-type inequality, and a high-dimensional Expander Mixing Lemma. As a corollary, using the work of Pach, we obtain a connection between spectral properties of complexes an…
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 …
We give a combinatorial proof of the result of Kahn, Kalai, and Linial, which states that every balanced boolean function on the $n$-dimensional boolean cube has a variable with influence of at least Omega(\frac{log n}{n}). The methods of the proof are then used to recover additional isoperimetric results for the cube, with improved constants. We also state some conjectures about optimal constants and discuss their possible implications
On modular inequalities for plane projective curves
本文通过组合Aomoto复形构造引入平面曲线补集的模不等式，用于研究扭曲Alexander多项式，特别是Milnor纤维的特征多项式，并给出正特征域上共振非平凡性的判据及根重数下界。
The study of ordering polytopes has been essential to the solution of various challenging combinatorial optimization problems. For instance, the incorporation of facet defining inequalities (FDIs) from these polytopes in branch-and-cut approaches represents among the most effective solution methodologies known to date for some of these problems. The weak order polytope, defined as the convex hull of the characteristic vectors of all binary orders on $n$ alternatives that are reflexive, transitive, and complete, has been particularly important for tackling problems in computational social choic…
From a combinatorial perspective, we establish three inequalities on coefficients of $R$- and Kazhdan-Lusztig polynomials for crystallographic Coxeter groups: (1) Nonnegativity of $(q-1)$-coefficients of $R$-polynomials, (2) a new criterion of rational singularities of Bruhat intervals by sum of quadratic coefficients of $R$-polynomials, (3) existence of a certain strict inequality (coefficientwise) of Kazhdan-Lusztig polynomials. Our main idea is to understand Deodhar's inequality in a connection with a sum of $R$-polynomials and edges of Bruhat graphs.
In order to provide a unified combinatorial interpretation of congruences modulo $5$ for 2-colored partition functions, Garvan introduced a bicrank statistic in terms of weighted vector partitions. In this paper, we obtain some inequalities between the bicrank counts $M^{*}(r,m,n)$ for $m=2$, $3$ and $4$ via their asymptotic formulas and some $q$-series techniques. These inequalities are parallel to Andrews and Lewis' results on the rank and crank counts for ordinary partitions.
On universal quadratic inequalities for minors of TNN matrices
本研究给出了所有n×n'全非负（TNN）矩阵均成立的子式二次不等式集合的组合刻画，该结论由研究者此前通过Lindström构造得到的平面图流生成矩阵子式的稳定二次恒等式结果推导而来。
Topological inequalities for conic-line arrangements with some prescribed singularities
该研究从4维流形视角引入组合二次曲线-直线构形，证明了ADE型偶总次数构形的Hirzebruch型不等式，还通过相关理论得到可分与奇数型构形消解重数序列的不等式与同余式。
Correlation inequalities for linear extensions
该研究采用组合图谱技术证明有限偏序集线性扩张数的新相关不等式（含与Stanley不等式相关的近似独立性结果），并拓展应用至标准杨表数和欧拉数领域。
Combinatorial Bounds for List Recovery via Discrete Brascamp--Lieb Inequalities
本文利用离散熵Brascamp--Lieb不等式，为多类线性码和折叠线性码的列表恢复问题建立了新的组合界，解决了列表大小能否被ℓ的多项式界定的长期开放问题。
In this paper, we provide a combinatorial/numerical method to establish new hypercontractivity estimates in group von Neumann algebras. We will illustrate our method with free groups, triangular groups and finite cyclic groups, for which we shall obtain optimal time hypercontractive $L_2 \to L_q$ inequalities with respect to the Markov process given by the word length and with $q$ an even integer. Interpolation and differentiation also yield general $L_p \to L_q$ hypercontrativity for $1 < p \le q < \infty$ via logarithmic Sobolev inequalities. Our method admits further applications to other d…
Many problems in combinatorial linear algebra require upper bounds on the number of solutions to an underdetermined system of linear equations $Ax = b$, where the coordinates of the vector $x$ are restricted to take values in some small subset (e.g. $\{\pm 1\}$) of the underlying field. The classical ways of bounding this quantity are to use either a rank bound observation due to Odlyzko or a vector anti-concentration inequality due to Halász. The former gives a stronger conclusion except when the number of equations is significantly smaller than the number of variables; even in such situation…
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