Richard P. Stanley
A survey of three recent developments in algebraic combinatorics: (1) the Laurent phenomenon, (2) Gromov-Witten invariants and toric Schur functions, and (3) toric h-vectors and intersection cohomology. This paper is a continuation of "Recent progress in algebraic combinatorics" (math.CO/0010218), which dealt with three other topics.
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A survey written for the upcoming "Handbook of Enumerative Combinatorics".
Lecture notes on algebraic methods in combinatorics
本研究为2023年春莱比锡课程的讲义，涵盖极值组合学、代数方法及组合几何，内容非详尽且未收录课程中的多数证明。
A survey of recent progress in three areas of algebraic combinatorics: (1) the Saturation Conjecture for Littlewood-Richardson coefficients, (2) the n! and (n+1)^{n-1} conjectures, and (3) longest increasing subsequences of permutations.
We consider the algebraic combinatorics of the set of injections from a $k$-element set to an $n$-element set. In particular, we give a new combinatorial formula for the spherical functions of the Gelfand pair $(S_k \times S_n, \text{diag}(S_k) \times S_{n-k})$. We use this combinatorial formula to give new Delsarte linear programming bounds on the size of codes over injections.
We describe how to construct and enumerate Magic squares, Franklin squares, Magic cubes, and Magic graphs as lattice points inside polyhedral cones using techniques from Algebraic Combinatorics. The main tools of our methods are the Hilbert Poincare series to enumerate lattice points and the Hilbert bases to generate lattice points. We define polytopes of magic labelings of graphs and digraphs, and give a description of the faces of the Birkhoff polytope as polytopes of magic labelings of digraphs.
Some applications of linear algebraic methods in combinatorics
本文将由小交易生成拉丁方及构造4-循环系统的问题转化为求特殊矩阵的核的基，并结合图结构展示线性代数方法在组合数学中的应用。
We formulate a number of new results in Algebraic Geometry and outline their derivation from Theorem 2.12 which belongs to Algebraic Combinatorics.
Short Proofs in Algebraic and Enumerative Combinatorics
本文利用ChatGPT 5.4 Pro自主生成简短证明，解决了模格上的梯队运动算子、停车函数统计以及plactic幺半群中心化子等领域的若干开放问题。
An Introduction to Algebraic Combinatorics
本文为研究生课程编写的代数组合学导论，系统介绍了形式幂级数、整数分拆、排列、减法方法及对称多项式，并证明了Jacobi三重积恒等式与Littlewood--Richardson法则，附录含200余道习题。
A partly autobiographical survey of the development of enumerative and algebraic combinatorics in the 1960's and 1970's.
Snakes are analogues of alternating permutations defined for any Coxeter group. We study these objects from the point of view of combinatorial Hopf algebras, such as noncommutative symmetric functions and their generalizations. The main purpose is to show that several properties of the generating functions of snakes, such as differential equations or closed form as trigonometric functions, can be lifted at the level of noncommutative symmetric functions or free quasi-symmetric functions. The results take the form of algebraic identities for type B noncommutative symmetric functions, noncommuta…
A polynomial has saturated Newton polytope (SNP) if every lattice point of the convex hull of its exponent vectors corresponds to a monomial. We compile instances of SNP in algebraic combinatorics (some with proofs, others conjecturally): skew Schur polynomials; symmetric polynomials associated to reduced words, Redfield--Polya theory, Witt vectors, and totally nonnegative matrices; resultants; discriminants (up to quartics); Macdonald polynomials; key polynomials; Demazure atoms; Schubert polynomials; and Grothendieck polynomials, among others. Our principal construction is the Schubitope. Fo…
Partially commutative monoids provide a powerful tool to study graphs, viewingwalks as words whose letters, the edges of the graph, obey a specific commutation rule. A particularclass of traces emerges from this framework, the hikes, whose alphabet is the set of simple cycleson the graph. We show that hikes characterize undirected graphs uniquely, up to isomorphism, andsatisfy remarkable algebraic properties such as the existence and uniqueness of a prime factorization.Because of this, the set of hikes partially ordered by divisibility hosts a plethora of relations in directcorrespondence with…
New classes of groups related to algebraic combinatorics with applications to isomorphism problems
本文引入$5/2$-群类的两个精化，证明其正则循环子群共轭，从而推广并简化了对称构型与单位循环有向图同构的判定证明。
Towards inductive proofs in algebraic combinatorics
本文引入了一类新的传递置换群，给出了其商保持在该族中的充分条件，并利用归纳法确定了阶为$p^n$的循环有向图的自同构群。
This is a survey of recent developments in combinatorics. The goal is to give a big picture of its many interactions with other areas of mathematics, such as: group theory, representation theory, commutative algebra, geometry (including algebraic geometry), topology, probability theory, and theoretical computer science.
We explain connections among several, a priori unrelated, areas of mathematics: combinatorics, algebraic statistics, topology and enumerative algebraic geometry. Our focus is on discrete invariants, strongly related to the theory of Lorentzian polynomials. The main concept joining the mentioned fields is a linear space of matrices.
Formalising Fisher's Inequality: Formal Linear Algebraic Proof Techniques in Combinatorics
本文提出了在Isabelle/HOL中用于关联结构证明的形式化线性代数技术，首次完成了Fisher不等式的形式化，并展示了该技术可适应组合学的未来形式化工作。
Signed combinatorial interpretations in algebraic combinatorics
本文证明了代数组合学中几大类结构常数（包括对称、拟对称多项式基及Schubert理论基）存在带符号组合解释，结果以计算复杂性语言表述，证明基于有效的Möbius反演。
We obtain a necessary and sufficient condition for the linear independence of solutions of differential equations for hyperlogarithms. The key fact is that the multiplier (i.e. the factor $M$ in the differential equation $dS=MS$) has only singularities of first order (Fuchsian-type equations) and this implies that they freely span a space which contains no primitive. We give direct applications where we extend the property of linear independence to the largest known ring of coefficients.
Arithmetic combinatorics is often concerned with the problem of bounding the behaviour of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry is often concerned with the problem of bounding the behaviour of arbitrary finite collections of geometric objects such as points, lines, or circles with respect to geometric operations such as incidence or distance. Given the presence of arbitrary finite sets in these problems, the methods used to attack these problems have primarily been combinatorial in na…
We introduce a new algebraic construction, {\em monop}, that combines monoids (with respect to the product of species), and operads (monoids with respect to the substitution of species) in the same algebraic structure. By the use of properties of cancellative set-monops we construct a family of partially ordered sets whose prototypical examples are the Dowling lattices. They generalize the partition posets associated to a cancellative operad, and the subset posets associated to a cancellative monoid. Their generalized Withney numbers of the first and second kind are the entries of a Riordan ma…
This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings, kissing numbers, lattices, and association schemes. In particular, special interest is devoted to the relationship between codes and combinatorial designs. We describe and recapitulate important results in the development of the state of the art. In addition, we give illustrative examples and constructions, and highlight recent advances. Finally, we provide…
This was the basis of two lectures in the Current Developments in Mathematics conference in 2011. These lectures survey the theory of hyperbolic and stable polynomials, from their origins in the theory of linear PDE's to their present uses in combinatorics and probability theory.
We give a brief account of a construction called tokens here, which is significant in algebra, analysis, combinatorics, and physics. Tokens allow to express a semigroup on one set via a semigroup convolution on another set. Therefore tokens are similar to intertwining operators but are more flexible. Keywords: semigroups, hypergroups, tokens, poset, multiplicative functions, polynomial sequence of binomial type, integral kernel, wavelets, refinement equation, special functions, quantum propagator, path integral, quantum computing.
Many Riordan arrays play a significant role in algebraic combinatorics. We explore the inversion of Riordan arrays in this context. We give a general construct for the inversion of a Riordan array, and study this in the case of various subgroups of the Riordan group. For instance, we show that the inversion of an ordinary Bell matrix is an exponential Riordan array in the associated subgroup. Examples from combinatorics and algebraic combinatorics illustrate the usefulness of such inversions. We end with a brief look at the inversion of exponential Riordan arrays. A final example places Airey'…
This paper surveys some combinatorial aspects of Smith normal form, and more generally, diagonal form. The discussion includes general algebraic properties and interpretations of Smith normal form, critical groups of graphs, and Smith normal form of random integer matrices. We then give some examples of Smith normal form and diagonal form arising from (1) symmetric functions, (2) a result of Carlitz, Roselle, and Scoville, and (3) the Varchenko matrix of a hyperplane arrangement.
We show, in great detail, how the perturbative tools of quantum field theory allow one to rigorously obtain: a ``categorified'' Faa di Bruno type formula for multiple composition, an explicit formula for reversion and a proof of Lagrange-Good inversion, all in the setting of multivariable power series. We took great pains to offer a self-contained presentation that, we hope, will provide any mathematician who wishes, an easy access to the wonderland of quantum field theory.
Fock and Goncharov introduced a family of cluster algebras associated with the moduli of SL(k)-local systems on a marked surface with extra decorations at marked points. We study this family from an algebraic and combinatorial perspective, emphasizing the structures which arise when the surface has punctures. When k is 2, these structures are the tagged arcs and tagged triangulations of Fomin, Shapiro, and Thurston. For higher k, the tagging of arcs is replaced by a Weyl group action at punctures discovered by Goncharov and Shen. We pursue a higher analogue of a tagged triangulation in the lan…
The Gini index is a number that attempts to measure how equitably a resource is distributed throughout a population, and is commonly used in economics as a measurement of inequality of wealth or income. The Gini index is often defined as the area between the "Lorenz curve" of a distribution and the line of equality, normalized to be between zero and one. In this fashion, we will define a Gini index on the set of integer partitions and prove some combinatorial results related to it; culminating in the proof of an identity for the expected value of the Gini index. We will then discuss symmetric …
We present a beautiful interplay between combinatorial topology and homological algebra for a class of monoids that arise naturally in algebraic combinatorics. We explore several applications of this interplay. For instance, we provide a new interpretation of the Leray number of a clique complex in terms of non-commutative algebra. Résumé. Nous présentons une magnifique interaction entre la topologie combinatoire et l'algèbre homologique d'une classe de monoïdes qui figurent naturellement dans la combinatoire algébrique. Nous explorons plusieurs applications de cette interaction. Par exemple, …
Operads are algebraic devices offering a formalization of the concept of operations with several inputs and one output. Such operations can be naturally composed to form bigger and more complex ones. Coming historically from algebraic topology, operads intervene now as important objects in computer science and in combinatorics. The theory of operads, together with the algebraic setting and the tools accompanying it, promises advances in these two areas. On the one hand, operads provide a useful abstraction of formal expressions, and also, provide connections with the theory of rewrite systems.…
The antiprism triangulation provides a natural way to subdivide a simplicial complex $Δ$, similar to barycentric subdivision, which appeared independently in combinatorial algebraic topology and computer science. It can be defined as the simplicial complex of chains of multi-pointed faces of $Δ$, from a combinatorial point of view, and by successively applying the antiprism construction, or balanced stellar subdivisions, on the faces of $Δ$, from a geometric point of view. This paper studies enumerative invariants associated to this triangulation, such as the transformation of the $h$-vector o…
The main ideas developed in this habilitation thesis consist in endowing combinatorial objects (words, permutations, trees, Young tableaux, etc.) with operations in order to construct algebraic structures. This process allows, by studying algebraically the structures thus obtained (changes of bases, generating sets, presentations, morphisms, representations), to collect combinatorial information about the underlying objects. The algebraic structures the most encountered here are magmas, posets, associative algebras, dendriform algebras, Hopf bialgebras, operads, and pros. This work explores th…
We give a coalgebra structure on 1-vertex irreducible graphs which is that of a cocommutative coassociative graded connected coalgebra. We generalize the coproduct to the algebraic representation of graphs so as to express a bare 1-particle irreducible n-point function in terms of its loop order contributions. The algebraic representation is so that graphs can be evaluated as Feynman graphs.
Machine Checked Proofs and Programs in Algebraic Combinatorics
本文介绍一个基于 Coq/Rocq 与 Mathematical Components 的代数组合学形式化库，核心贡献是机器检验的 Littlewood-Richardson 法则证明及相关认证算法实现。
We survey research relating algebraic properties of powers of squarefree monomial ideals to combinatorial structures. In particular, we describe how to detect important properties of (hyper)graphs by solving ideal membership problems and computing associated primes. This work leads to algebraic characterizations of perfect graphs independent of the Strong Perfect Graph Theorem. In addition, we discuss the equivalence between the Conforti-Cornuejols conjecture from linear programming and the question of when symbolic and ordinary powers of squarefree monomial ideals coincide.
We consider generators of algebraic covariant derivative curvature tensors R' which can be constructed by a Young symmetrization of product tensors W*U or U*W, where W and U are covariant tensors of order 2 and 3. W is a symmetric or alternating tensor whereas U belongs to a class of the infinite set S of irreducible symmetry classes characterized by the partition (2,1). Using Computer Algebra we search for such generators whose coordinate representations are polynomials with a minimal number of summands. For a generic choice of the symmetry class of U we obtain lengths of 16 or 20 summands if…
Integrability and combinatorics
本文探讨了将可积系统方法应用于组合学问题，并以交错符号矩阵枚举和对称多项式理论为例进行了阐述。
Additive combinatorics is built around the famous theorem by Szemerédi which asserts existence of arithmetic progressions of any length among the integers. There exist several different proofs of the theorem based on very different techniques. Szemerédi's theorem is an existence statement, whereas the ultimate goal in combinatorics is always to make enumeration statements. In this article we develop new methods based on real algebraic geometry to obtain several quantitative statements on the number of arithmetic progressions in fixed density sets. We further discuss the possibility of a genera…
The stabilization algorithm of Weisfeiler and Leman has as an input any square matrix A of order n and returns the minimal cellular (coherent) algebra W(A) which includes A. In case when A=A(G) is the adjacency matrix of a graph G the algorithm examines all configurations in G having three vertices and, according to this information, partitions vertices and ordered pairs of vertices into equivalence classes. The resulting construction allows to associate to each graph G a matrix algebra W(G):= W(A(G))$ which is an invariant of the graph G. For many classes of graphs, in particular for most of …
The interaction between combinatorics and algebraic and differential geometry is very strong. While researching a problem of Hessian topology, we came across a series of identities of binomial coefficients, which are useful for proving a topological property of certain spaces whose elements are graphs of a class of hyperbolic polynomials. These identities are proven by different methods in combinatorics.
Machine Learning meets Algebraic Combinatorics: A Suite of Datasets Capturing Research-level Conjecturing Ability in Pure Mathematics
本文提出代数组合学数据集仓库（ACD Repo），包含九个研究级开放问题及多达千万级示例，旨在支持机器学习模型从数据中生成数学猜想，并探讨了窄模型训练、可解释性分析和LLM程序合成等应用方式及数据集设计挑战。
The classical theory of symmetric functions has a central position in algebraic combinatorics, bridging aspects of representation theory, combinatorics, and enumerative geometry. More recently, this theory has been fruitfully extended to the larger ring of quasisymmetric functions, with corresponding applications. Here, we survey recent work extending this theory further to general asymmetric polynomials.
We discuss three distinct topics of independent interest; one in enumerative combinatorics, one in symmetric function theory, and one in algebraic geometry. The topic in enumerative combinatorics concerns a q-analog of a generalization of the Eulerian polynomials, the one in symmetric function theory deals with a refinement of the chromatic symmetric functions of Stanley, and the one in algebraic geometry deals with Tymoczko's representation of the symmetric group on the cohomology of the regular semisimple Hessenberg variety of type A. Our purpose is to explore some remarkable connections bet…
We study some combinatorial and algebraic properties of certain quadratic algebras related with dynamical classical and classical Yang-Baxter equations. One can find more details about the content of present paper in Extended Abstract.
In this note we list a number of open problems in the fields of number theory, combinatorics, and representation theory: algebraic functions with Fermat property; power product expansion of the generating function for the partition function; relation between the non-trivial Riemann zeros and eigenvalues of the transfer operator; functional equation related to norm forms; two problems from geometric combinatorics; a problem on the moments of the Minkowski question mark function; a question in representation theory; a problem on interpolating the moments of the Stern's diatomic sequence; an arit…
We give an overview of recent progress around a problem introduced by Elekes and Rónyai. The prototype problem is to show that a polynomial $f\in \mathbb{R}[x,y]$ has a large image on a Cartesian product $A\times B\subset \mathbb{R}^2$, unless $f$ has a group-related special form. We discuss a number of variants and generalizations. This includes the Elekes-Szabó problem, which generalizes the Elekes-Rónyai problem to a question about an upper bound on the intersection of an algebraic surface with a Cartesian product, and curve variants, where we ask the same questions for Cartesian products o…
We reconsider the classical equality 0.999. .. = 1 with the tool of circular words, that is: finite words whose last letter is assumed to be followed by the first one. Such circular words are naturally embedded with algebraic structures that enlight this problematic equality, allowing it to be considered in Q rather than in R. We comment early history of such structures, that involves English teachers and accountants of the first part of the xviii th century, who appear to be the firsts to assert the equality 0.999. .. = 1. Their level of understanding show links with Dubinsky et al.'s apos th…
Computational Complexity in Algebraic Combinatorics
本文探讨代数组合学优美乘积公式（如Kronecker和plethysm系数）的计算复杂性，并指出该研究推翻了Geometric Complexity Theory解决VP vs VNP问题的设想。
The algebraic and combinatorial theory of shuffles, introduced by Chen and Ree, is further developed and applied to the study of multiple zeta values. In particular, we establish evaluations for certain sums of cyclically generated multiple zeta values. The boundary case of our result reduces to a former conjecture of Zagier.
PermuFormer: Multi-Task Pretraining for Permutation Representation in Algebraic Combinatorics
本文提出PermuFormer，一个在28亿标记多任务语料上预训练的自回归Transformer，用于代数组合学排列任务，微调后优于从零训练模型和通用语言模型，并分析了其内部解码机制。
We demonstrate the use of several tools from Algebraic Combinatorics such as Young tableaux, symmetry operators, the Littlewood-Richardson rule and discrete Fourier transforms of symmetric groups in investigations of algebraic curvature tensors.
We survey the relationship between the combinatorics and geometry of graphs and the algebraic structure of right-angled Artin groups. We concentrate on the defining graph of the right-angled Artin group and on the extension graph associated to the right-angled Artin group. Additionally, we discuss connections to geometric group theory and complexity theory. The final version of this survey will appear in "In the tradition of Thurston, vol.~II", ed.~K.~Ohshika and A.~Papadopoulos.
A SageMath Package for Analytic Combinatorics in Several Variables: Beyond the Smooth Case
本文提出首个基于SageMath的软件，通过结合Whitney分层与ACSV方法，严格计算生成函数具有非光滑奇点簇的多元序列渐近性，并改进了高阶渐近展开与临界点计算。
We study some algebraic and combinatorial features of two algebras that arise as quotients of Temperley-Lieb algebras of type $\tilde{C}$, namely, the two-boundary Temperley-Lieb algebra and the symplectic blob algebra. We provide a monomial basis for both algebras. The elements of these bases are parameterized by certain subsets of fully commutative elements. We enumerate these elements according to their affine length.
George Birkhoff proved in 1912 that the number of proper colorings of a finite graph G with n colors is a polynomial in n, called the chromatic polynomial of G. Read conjectured in 1968 that for any graph G, the sequence of absolute values of coefficients of the chromatic polynomial is unimodal: it goes up, hits a peak, and then goes down. Read's conjecture was proved by June Huh in a 2012 paper making heavy use of methods from algebraic geometry. Huh's result was subsequently refined and generalized by Huh and Katz, again using substantial doses of algebraic geometry. Both papers in fact esta…
On complex algebraic varieties, height functions arising in combinatorial applications fail to be proper. This complicates the description and computation via Morse theory of key topological invariants. Here we establish checkable conditions under which the behavior at infinity may be ignored, and the usual theorems of classical and stratified Morse theory may be applied. This allows for simplified arguments in the field of analytic combinatorics in several variables, and forms the basis for new methods applying to problems beyond the reach of previous techniques.
Using standard techniques from combinatorics, model theory, and algebraic geometry, we prove generalized versions of several basic results in the theory of spectrally arbitrary matrix patterns. Also, we point out a counterexample to a conjecture proposed recently by McDonald and Melvin.
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…
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