Proof of two conjectures on correlation inequalities for one class of monotone functions

A Proof of the Strict Monotone 5-step Conjecture

A computer search through the oriented matroid programs with dimension 5 and 10 facets shows that the maximum strictly monotone diameter is 5. Thus $Δ_{sm}(5,10)=5$. This enumeration is analogous to that of Bremner and Schewe for the non-monotone diameter of 6-polytopes with 12 facets. Similar enumerations show that $Δ_{sm}(4,9)=5$ and $Δ_m(4,9)=Δ_m(5,10)=6.$ We shorten the known non-computer proof of the strict monotone 4-step conjecture.

Influences of monotone Boolean functions

Recently, Keller and Pilpel conjectured that the influence of a monotone Boolean function does not decrease if we apply to it an invertible linear transformation. Our aim in this short note is to prove this conjecture.

关于吕与张的十四个等分布猜想及带角阴影的单调网格模式

On fourteen equidistribution conjectures of Lv and Zhang and monotone mesh patterns with corner shadings

本文通过构造三个类似补集的排列映射，证明并推广了吕与张的全部十四个联合对称等分布猜想，并给出了角阴影受限的若干网格模式避免排列的计数结果。

Topological recursion for monotone orbifold Hurwitz numbers: a proof of the Do-Karev conjecture

We prove the conjecture of Do and Karev that the monotone orbifold Hurwitz numbers satisfy the Chekhov-Eynard-Orantin topological recursion.

Proof of the monotone column permanent conjecture

Let A be an n-by-n matrix of real numbers which are weakly decreasing down each column, Z_n = diag(z_1,..., z_n) a diagonal matrix of indeterminates, and J_n the n-by-n matrix of all ones. We prove that per(J_nZ_n+A) is stable in the z_i, resolving a recent conjecture of Haglund and Visontai. This immediately implies that per(zJ_n+A) is a polynomial in z with only real roots, an open conjecture of Haglund, Ono, and Wagner from 1999. Other applications include a multivariate stable Eulerian polynomial, a new proof of Grace's apolarity theorem and new permanental inequalities.

Chvátal's Conjecture and Correlation Inequalities

Chvátal's conjecture in extremal combinatorics asserts that for any decreasing family $\mathcal{F}$ of subsets of a finite set $S$, there is a largest intersecting subfamily of $\mathcal{F}$ consisting of all members of $\mathcal{F}$ that include a particular $x \in S$. In this paper we reformulate the conjecture in terms of influences of variables on Boolean functions and correlation inequalities, and study special cases and variants using tools from discrete Fourier analysis.