Combinatorics of geometrically distributed random variables: Value and position of the r-th left-to-right maximum

Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry

In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent paper, the authors established a close conn…

Combinatorial Topology and the Global Dimension of Algebras Arising in Combinatorics

In a highly influential paper, Bidigare, Hanlon and Rockmore showed that a number of popular Markov chains are random walks on the faces of a hyperplane arrangement. Their analysis of these Markov chains took advantage of the monoid structure on the set of faces. This theory was later extended by Brown to a larger class of monoids called left regular bands. In both cases, the representation theory of these monoids played a prominent role. In particular, it was used to compute the spectrum of the transition operators of the Markov chains and to prove diagonalizability of the transition operator…

组合学中的次乘法多项式

Submultiplicative Polynomials in Combinatorics

研究针对归一化序列递归定义的多项式\(P_n^g(x)\)，探讨其次乘法性质，给出建立该性质的有效准则。

Tableaux combinatorics for the asymmetric exclusion process

The partially asymmetric exclusion process (PASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of $n$ sites. It is partially asymmetric in the sense that the probability of hopping left is $q$ times the probability of hopping right. Additionally, particles may enter from the left with probability $α$ and exit from the right with probability $β$. In this paper we prove a close connection between the PASEP and the combinatorics of permutation tableaux. (These tableaux come indirectly from t…

具有相等左右商集的有限群的分类

Classification of Finite Groups With Equal Left and Right Quotient Sets

本文分类了所有满足任意子集左右商集基数相等的有限群，并回应了相关加性组合问题。

On the maximum number of maximum independent sets in connected graphs

We characterize the connected graphs of given order $n$ and given independence number $α$ that maximize the number of maximum independent sets. For $3\leq α\leq n/2$, there is a unique such graph that arises from the disjoint union of $α$ cliques of orders $\left\lceil\frac{n}α\right\rceil$ and $\left\lfloor\frac{n}α\right\rfloor$, by selecting a vertex $x$ in a largest clique and adding an edge between $x$ and a vertex in each of the remaining $α-1$ cliques. Our result confirms a conjecture of Derikvand and Oboudi [On the number of maximum independent sets of graphs, Transactions on Combinato…

关于斐波那契密度上无限词 $\mathfrak{F}=\mathfrak{F}_{b}:=\left({ }_{b} f_{n}\right)_{n \geqslant 0}$ 的改进遍历理论

Improvement Ergodic Theory For The Infinite Word $\mathfrak{F}=\mathfrak{F}_{b}:=\left({ }_{b} f_{n}\right)_{n \geqslant 0}$ on Fibonacci Density

本文研究斐波那契密度上无限词的遍历理论，分析其子词多样性与广义模式分布，并揭示替换规则、代数数论与组合复杂性之间的深层联系。

Combinatorics of the asymmetric exclusion process on a semi-infinite lattice

We study two versions of the asymmetric exclusion process (ASEP) -- an ASEP on a semi-infinite lattice with an open left boundary, and an ASEP on a finite lattice with open left and right boundaries -- and we demonstrate a surprising relationship between their stationary measures. The semi-infinite ASEP was first studied by Liggett and then Grosskinsky, while the finite ASEP had been introduced earlier by Spitzer and Macdonald-Gibbs-Pipkin. We show that the finite correlation functions involving the first L sites for the stationary measures on the semi-infinite ASEP can be obtained as a nonphy…

Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials

Introduced in the late 1960's, the asymmetric exclusion process (ASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites with open boundaries. It has been cited as a model for traffic flow and protein synthesis. In the most general form of the ASEP with open boundaries, particles may enter and exit at the left with probabilities alpha and gamma, and they may exit and enter at the right with probabilities beta and delta. In the bulk, the probability of hopping left is q times the prob…

Extensions of Autocorrelation Inequalities with Applications to Additive Combinatorics

In a 2019 paper, Barnard and Steinerberger show that for $f\in L^1(\mathbf{R})$, the following autocorrelation inequality holds: \begin{equation*} \min_{0 \leq t \leq 1} \int_\mathbf{R} f(x) f(x+t)\ \mathrm{d}x \ \leq\ 0.411 ||f||_{L^1}^2, \end{equation*} where the constant $0.411$ cannot be replaced by $0.37$. In addition to being interesting and important in their own right, inequalities such as these have applications in additive combinatorics where some problems, such as those of minimal difference basis, can be encapsulated by a convolution inequality similar to the above integral. Barnar…

An upper bound on Pachner moves relating geometric triangulations

We show that any two geometric triangulations of a closed hyperbolic, spherical or Euclidean manifold are related by a sequence of Pachner moves and barycentric subdivisions of bounded length. This bound is in terms of the dimension of the manifold, the number of top dimensional simplexes and bound on the lengths of edges of the triangulation. This leads to an algorithm to check from the combinatorics of the triangulation and bounds on lengths of edges, if two geometrically triangulated closed hyperbolic or low dimensional spherical manifolds are isometric or not.

两类广义Fibonacci子序列的渐近估计

The asymptotic estimation for two classes of generalized Fibonacci sub-sequences

本文利用初等方法，对广义Fibonacci序列的两类无穷级数倒数的渐近估计进行了研究，推广了Yuan等人2025年的结果。

Log-Concavity of Infinite Product and Infinite Sum Generating Functions

We expand on the remark by Andrews on the importance of infinite sums and products in combinatorics. Let $\{g_d(n)\}_{d\geq 0,n \geq 1}$ be the double sequences $σ_d(n)= \sum_{\ell \mid n} \ell^d$ or $ψ_d(n)= n^d$. We associate double sequences $\left\{ p^{g_{d} }\left( n\right) \right\}$ and $\left\{ q^{g_{d} }\left( n\right) \right\} $, defined as the coefficients of \begin{eqnarray*} \sum_{n=0}^{\infty} p^{g_{d} }\left( n\right) \, t^{n} & := & \prod_{n=1}^{\infty} \left( 1 - t^{n} \right)^{-\frac{ \sum_{\ell \mid n} μ(\ell) \, g_d(n/\ell) }{n} }, \\ \sum_{n=0}^{\infty} q^{g_{d} }\left( n\r…

Some congruences involving binomial coefficients

Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let $p>3$ be a prime. We show that $$T_{p-1}\equiv\left(\frac p3\right)3^{p-1}\ \pmod{p^2},$$ where the central trinomial coefficient $T_n$ is the constant term in the expansion of $(1+x+x^{-1})^n$. We also prove three congruences modulo $p^3$ conjectured by Sun, one of which is $$\sum_{k=0}^{p-1}\binom{p-1}k\binom{2k}k((-1)^k-(-3)^{-k})\equiv \left(\frac p3\right)(3^{p-1}-1)\ \pmod{p^3}.$$ In addition, we get some new combinatorial identities.

大整环中加法与乘法中心集的单色性及 Goswami 定理

Monocromaticity of additive and multiplicative central sets and Goswami's theorem in large Integral Domains

本文将 Goswami 关于素数集乘积包含 k·N 的结果推广到大整环，证明组合丰富集 A 满足 k·N⊂(A−A)·(A−A)，并首次证明大整环有限划分中至少一个单元同时为加法与乘法中心集及其逆命题。

群上具有单位积的项链

Necklaces over a group with identity product

该研究探讨了群上具有单位积的项链计数问题，证明两种不同计数方式结果相同，并将其推广至更一般的项链计数场景。

Distributions on partitions arising from Hilbert schemes and hook lengths

Recent works at the interface of algebraic combinatorics, algebraic geometry, number theory, and topology have provided new integer-valued invariants on integer partitions. It is natural to consider the distribution of partitions when sorted by these invariants in congruence classes. We consider the prominent situations which arise from extensions of the Nekrasov-Okounkov hook product formula, and from Betti numbers of various Hilbert schemes of $n$ points on $\mathbb{C}^2.$ For the Hilbert schemes, we prove that homology is equidistributed as $n\to \infty.$ For $t$-hooks, we prove distributio…

分裂素数与Elekes-Rónyai问题

Split primes and the Elekes-Rónyai problem

构造反例表明存在绝对常数c>0和任意大的有限实数集A，使得多项式x+y+(x-y)^2的值集大小不超过|A|^{2-c}，从而否定Elekes-Rónyai猜想。

Near-Optimal Induced Universal Graphs for Bounded Degree Graphs

A graph $U$ is an induced universal graph for a family $F$ of graphs if every graph in $F$ is a vertex-induced subgraph of $U$. For the family of all undirected graphs on $n$ vertices Alstrup, Kaplan, Thorup, and Zwick [STOC 2015] give an induced universal graph with $O\!\left(2^{n/2}\right)$ vertices, matching a lower bound by Moon [Proc. Glasgow Math. Assoc. 1965]. Let $k= \lceil D/2 \rceil$. Improving asymptotically on previous results by Butler [Graphs and Combinatorics 2009] and Esperet, Arnaud and Ochem [IPL 2008], we give an induced universal graph with $O\!\left(\frac{k2^k}{k!}n^k \rig…

Pointwise densities of homogeneous Cantor measure and critical values

Let $N\ge 2$ and $ρ\in(0,1/N^2]$. The homogenous Cantor set $E$ is the self-similar set generated by the iterated function system \[ \left\{f_i(x)=ρx+\frac{i(1-ρ)}{N-1}: i=0,1,\ldots, N-1\right\}. \] Let $s=\dim_H E$ be the Hausdorff dimension of $E$, and let $μ=\mathcal H^s|_E$ be the $s$-dimensional Hausdorff measure restricted to $E$. In this paper we describe, for each $x\in E$, the pointwise lower $s$-density $Θ_*^s(μ,x)$ and upper $s$-density $Θ^{*s}(μ, x)$ of $μ$ at $x$. This extends some early results of Feng et al. (2000). Furthermore, we determine two critical values $a_c$ and $b_c$ …

The strong circular law: a combinatorial view

Let $N_n$ be an $n\times n$ complex random matrix, each of whose entries is an independent copy of a centered complex random variable $z$ with finite non-zero variance $σ^{2}$. The strong circular law, proved by Tao and Vu, states that almost surely, as $n\to \infty$, the empirical spectral distribution of $N_n/(σ\sqrt{n})$ converges to the uniform distribution on the unit disc in $\mathbb{C}$. A crucial ingredient in the proof of Tao and Vu, which uses deep ideas from additive combinatorics, is controlling the lower tail of the least singular value of the random matrix $xI - N_{n}/(σ\sqrt{n})…

General Position Subsets and Independent Hyperplanes in d-Space

Erdős asked what is the maximum number $α(n)$ such that every set of $n$ points in the plane with no four on a line contains $α(n)$ points in general position. We consider variants of this question for $d$-dimensional point sets and generalize previously known bounds. In particular, we prove the following two results for fixed $d$: - Every set $H$ of $n$ hyperplanes in $\mathbb{R}^d$ contains a subset $S\subseteq H$ of size at least $c \left(n \log n\right)^{1/d}$, for some constant $c=c(d)>0$, such that no cell of the arrangement of $H$ is bounded by hyperplanes of $S$ only. - Every set of $c…