Balthazar Charles
In this extended abstract we announce a proof that, in a Coxeter group of rank 3, low elements are in bijection with small inversion sets. This gives a partial confirmation of Conjecture 2 in [Dyer, Hohlweg '16]. That same article provides the main ingredient: the bipodality of the set of small roots is used to propagate information on the vertices of inversion polytopes.
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In this article, we investigate the existence of joins in the weak order of an infinite Coxeter group W. We give a geometric characterization of the existence of a join for a subset X in W in terms of the inversion sets of its elements and their position relative to the imaginary cone. Finally, we discuss inversion sets of infinite reduced words and the notions of biconvex and biclosed sets of positive roots.
On the interaction of the Coxeter transformation and the rowmotion bijection
本文证明有限偏序集关联代数上rowmotion双射与Coxeter矩阵满足$(ρ^{-1}C)^2=id$，并将其推广到一般Auslander正则代数及高阶Auslander代数情形。
We prove that the restriction of Bruhat order to noncrossing partitions in type $A_n$ for the Coxeter element $c=s_1s_2 ...s_n$ forms a distributive lattice isomorphic to the order ideals of the root poset ordered by inclusion. Motivated by the change-of-basis from the graphical basis of the Temperley-Lieb algebra to the image of the simple elements of the dual braid monoid, we extend this bijection to other Coxeter elements using certain canonical factorizations. In particular, we give new bijections---fixing the set of reflections---between noncrossing partitions associated to distinct Coxet…
Finite Coxeter Groups and Generalized Elnitsky Tilings
本文基于Elnitsky对A、B、D型Coxeter群约化词与多边形铺砌的双射，通过嵌入对称群的方法，将其推广至所有有限Coxeter群。
Although both noncrossing partitions and nonnesting partitions are uniformly enumerated for Weyl groups, the exact relationship between these two sets of combinatorial objects remains frustratingly mysterious. In this paper, we give a precise combinatorial answer in the case of the symmetric group: for any standard Coxeter element, we construct an equivariant bijection between noncrossing partitions under the Kreweras complement and nonnesting partitions under a Coxeter-theoretically natural cyclic action we call the Kroweras complement. Our equivariant bijection is the unique bijection that i…
Coxeter systems, left inversion sets, and higher dimensional cubes
本文研究Coxeter系统中左反演集方程，定义并推广了Coxeter方块与高维Coxeter $n$-立方体，证明了其在有限群中的存在性，并建立了$A_n$系统中Coxeter $n$-立方体与二叉树及bigrassmannian排列的映射关系。
Auslander regular algebras and Coxeter matrices
研究奥斯特兰德正则代数与考克斯特矩阵，证明其等级双射与考克斯特矩阵布劳威尔分解中的置换矩阵\(P\)一致，给出新解释与快速计算方法，并应用于考克斯特矩阵行列式、分配格特征及半单李代数范畴\(\mathcal{O}\)的块中模的同调结果。
We investigate Mahonian and Eulerian probability distributions given by inversions and descents in general finite Coxeter groups. We provide uniform formulas for the means and variances in terms of Coxeter group data in both cases. We also provide uniform formulas for the double-Eulerian probability distribution of the sum of descents and inverse descents. We finally establish necessary and sufficient conditions for general sequences of Coxeter groups of increasing rank under which Mahonian and Eulerian probability distributions satisfy central and local limit theorems.
In Artin-Tits groups attached to Coxeter groups of spherical type, we give a combinatorial formula to express the simple elements of the dual braid monoids in the classical Artin generators. Every simple dual braid is obtained by lifting an $S$-reduced expression of its image in the Coxeter group, in a way which involves Reading's $c$-sortable elements. It has as an immediate consequence that simple dual braids are Mikado braids (the known proofs of this result either require topological realizations of the Artin groups or categorification techniques), and hence that their images in the Iwahor…
In this paper we introduce abstract string modules and give an explicit bijection between the submodule lattice of an abstract string module and the perfect matching lattice of the corresponding abstract snake graph. In particular, we make explicit the direct correspondence between a submodule of a string module and the perfect matching of the corresponding snake graph. For every string module, we define a Coxeter element in a symmetric group, and we establish a bijection between these lattices and the interval in the weak Bruhat order determined by the Coxeter element. Using the correspondenc…
A (semi)brick over an algebra $A$ is a module $S$ such that the endomorphism ring $\operatorname{\mathsf{End}}_A(S)$ is a (product of) division algebra. For each Dynkin diagram $Δ$, there is a bijection from the Coxeter group $W$ of type $Δ$ to the set of semibricks over the preprojective algebra $Π$ of type $Δ$, which is restricted to a bijection from the set of join-irreducible elements of $W$ to the set of bricks over $Π$. This paper is devoted to giving an explicit description of these bijections in the case $Δ=\mathbb{A}_n$ or $\mathbb{D}_n$. First, for each join-irreducible element $w \i…
In this article we introduce the notion of a \textit{regular partition} of a Coxeter group. We develop the theory of these partitions, and show that the class of regular partitions is essentially equivalent to the class of automata (not necessarily finite state) recognising the language of reduced words in the Coxeter group. As an application of this theory we prove that each cone type in a Coxeter group has a unique minimal length representative. This result can be seen as an analogue of Shi's classical result that each component of the Shi arrangement of an affine Coxeter group has a unique …
We provide a bijective proof of a formula of Auli and the author expressing the number of inversion sequences with no three consecutive equal entries in terms of the number of non-derangements, that is, permutations with fixed points. Additionally, we give bijective proofs of two simple recurrences for the number of non-derangements.
We introduce bijections between generalized type $A_n$ noncrossing partitions (that is, associated to arbitrary standard Coxeter elements) and fully commutative elements of the same type. The latter index the diagram basis of the classical Temperley-Lieb algebra, while for each choice of standard Coxeter element the corresponding noncrossing partitions also index a basis, given by the images in the Temperley-Lieb algebra of the simple elements of the dual Garside structure (associated to this choice of standard Coxeter element) of the Artin braid group on $n+1$ strands. We then show that our b…
We give a criterion for Bruhat order on noncrossing partitions corresponding to the Coxeter element $c=s_1 s_2\cdots s_n$. Using it we prove that the Bruhat order endows noncrossing partitions with a lattice structure. We then explain what happens if we change the Coxeter element; in that case the lattice property fails and we explain which order to consider to get the same lattice structure as for the Coxeter element $c$. In particular we get bijections between noncrossing partitions associated to distinct Coxeter elements, which fix the set of reflections.
In this paper we study the equivalence relation on the set of acyclic orientations of a graph Y that arises through source-to-sink conversions. This source-to-sink conversion encodes, e.g. conjugation of Coxeter elements of a Coxeter group. We give a direct proof of a recursion for the number of equivalence classes of this relation for an arbitrary graph Y using edge deletion and edge contraction of non-bridge edges. We conclude by showing how this result may also be obtained through an evaluation of the Tutte polynomial as T(Y,1,0), and we provide bijections to two other classes of acyclic or…
In their work on `Coxeter-like complexes', Babson and Reiner introduced a simplicial complex $Δ_T$ associated to each tree $T$ on $n$ nodes, generalizing chessboard complexes and type A Coxeter complexes. They conjectured that $Δ_T$ is $(n-b-1)$-connected when the tree has $b$ leaves. We provide a shelling for the $(n-b)$-skeleton of $Δ_T$, thereby proving this conjecture. In the process, we introduce notions of weak order and inversion functions on the labellings of a tree $T$ which imply shellability of $Δ_T$, and we construct such inversion functions for a large enough class of trees to ded…
Affine extended weak order is a lattice
本文证明了仿射类型Coxeter群中扩展弱序是一个完全格，通过引入清洁排列的概念并建立通用框架进行验证。
Set partitions avoiding $k$-crossing and $k$-nesting have been extensively studied from the aspects of both combinatorics and mathematical biology. By using the generating tree technique, the obstinate kernel method and Zeilberger's algorithm, Lin confirmed a conjecture due independently to the author and Martinez-Savage that asserts inversion sequences with no weakly decreasing subsequence of length 3 and enhanced 3-nonnesting partitions have the same cardinality. In this paper, we provide a bijective proof of this conjecture. Our bijection also enables us to provide a new bijective proof of …
It is difficult to identify a G-C fullerene directly from its dimensions as its lattice is not proportional to that of its archetype in general, although they have the same three-dimensional shape. In this paper, the area scale factor of a G-C fullerene is proved to be an integer, which can be calculated from its dimensions. All the G-C transformations are k-inflations that can be easily identified and inversed, primary transformations whose area scale factors are prime numbers, or composite transformations whose area scale factors are the product of those of its sub-transformations. As the re…
Bijections on pattern avoiding inversion sequences and related objects
本文通过引入 $F$-路径建立双射，统一刻画避免 $101$、$102$ 的逆序列及 Schröder 路径等等价计数对象，并研究相关统计量与直和。
Canonical Join Representations and Join-irreducible elements of Garside shadows in Coxeter groups
本文研究了Coxeter群中元素的规范连接表示和连接不可约元素的性质，并提出了确定元素锥型的高效计算方法。
We study the graph on reduced words with edges given by the Coxeter relations for the symmetric group. We define a metric on reduced words for a given permutation, analogous to Coxeter length for permutations, for which the graph becomes ranked with unique maximal element. We show this metric extends naturally to balanced tableaux, and use it to recover enumerative results of Edelman and Greene and of Reiner and Roichman.
We generalize two bijections due to Garsia and Gessel to compute the generating functions of the two vector statistics $(\des_G, \maj,\ell_G, \col)$ and $(\des_G, \ides_G, \maj, \imaj, \col, \icol)$ over the wreath product of a symmetric group by a cyclic group. Here $\des_G$, $\ell_G$, $\maj$, $\col$, $\ides_G$, $\imaj_G$, and $\icol$ denote the number of descents, length, major index, color weight, inverse descents, inverse major index, and inverse color weight, respectively. Our main formulas generalize and unify several known identities due to Brenti, Carlitz, Chow-Gessel, Garsia-Gessel, a…
Cover-inclusive Dyck tilings are tilings of skew Young diagrams with ribbon tiles shaped like Dyck paths, in which tiles are no larger than the tiles they cover. These tilings arise in the study of certain statistical physics models and also Kazhdan--Lusztig polynomials. We give two bijections between cover-inclusive Dyck tilings and linear extensions of tree posets. The first bijection maps the statistic (area + tiles)/2 to inversions of the linear extension, and the second bijection maps the "discrepancy" between the upper and lower boundary of the tiling to descents of the linear extension.
We generalize some identities and q-identities previously known for the symmetric group to Coxeter groups of type B and D. The extended results include theorems of Foata and Schützenberger, Gessel, and Roselle on various distributions of inversion number, major index, and descent number. In order to show our results we provide caracterizations of the systems of minimal coset representatives of Coxeter groups of type B and D.
We show that Han's bijection when restricted to permutations can be carried out in terms of the cyclic major code and the cyclic inversion code. In other words, it maps a permutation $π$ with a cyclic major code $(s_1, s_2, ..., s_n)$ to a permutation $σ$ with a cyclic inversion code $(s_1,s_2, ..., s_n)$. We also show that the fixed points of Han's map can be characterized by the strong fixed points of Foata's second fundamental transformation. The notion of strong fixed points is related to partial Foata maps introduced by Björner and Wachs.
Clusters, Coxeter-sortable elements and noncrossing partitions
本文研究Coxeter群中可排序元素与簇、非交叉划分的等价关系，通过双射证明其数量相等，并用反演集和经典案例中的排列特征进行刻画。
We consider the problem of finding a bijection between the sets of alternating sign matrices and of totally symmetric self complementary plane partitions, which can be reformulated using Gog and Magog triangles. In a previous work we introduced GOGAm triangles, which are images of Magog triangles by the Schützenberger involution. In this paper we introduce left Gog and GOGAm trapezoids. We conjecture that they are equienumerated, and we give an explicit bijection between such trapezoids with one or two diagonals. We also study the distribution of inversions and coinversions in Gog triangles.
We provide two explicit bijections demonstrating that, among permutations, the number of invisible inversions is equidistributed with the number of occurrences of the vincular pattern 13-2 after sorting the set of runs.
We present an explicit bijection between noncrossing and nonnesting partitions of Coxeter systems of type D which preserves openers, closers and transients.
Coxeter groups and Billey-Postnikov decompositions
本文综述了Billey-Postnikov分解在光滑Schubert簇分类计数、反演超平面排列及排列模式避免等领域的应用。
Stanley's formula for the number of reduced expressions of a permutation regarded as a Coxeter group element raises the question of how to enumerate the reduced expressions of an arbitrary Coxeter group element. We provide a framework for answering this question by constructing combinatorial objects that represent the inversion set and the reduced expressions for an arbitrary Coxeter group element. The framework also provides a formula for the length of an element formed by deleting a generator from a Coxeter group element. Fan and Hagiwara, et al$.$ showed that for certain Coxeter groups, the…
Bijections for generalized Wilf equivalences
本研究提出从组合恒等式的容斥证明出发、用递归减法构造直接双射的方法，将其用于逆序序列连续模式的广义Wilf等价研究，给出了已有结论的直接双射证明及更强模式关系的新双射证明。
An involution in a Coxeter group has an associated set of involution words, a variation on reduced words. These words are saturated chains in a partial order first considered by Richardson and Springer in their study of symmetric varieties. In the symmetric group, involution words can be enumerated in terms of tableaux using appropriate analogues of the symmetric functions introduced by Stanley to accomplish the same task for reduced words. We adapt this approach to the group of signed permutations. We show that involution words for the longest element in the Coxeter group $C_n$ are in bijecti…
On 102-avoiding inversion sequences
本文通过引入 UVD 路径和带标号 F-路径两类中间对象，建立了避开 102 模式的逆序序列与无峰无谷且以对角步结束的 2-Schröder 路径之间的双射，并基于自然统计量给出相应结构的枚举及精细化计数。
In a previous paper, we showed that $3\bar{5}241$-avoiding permutations are counted by the unique sequence that starts with a 1 and shifts left under the self-composition transform. The proof uses a complicated bijection. Here we give a much simpler proof based on Lagrange inversion.
On inversion sets of joins in weak Bruhat order
本文针对有限 Coxeter 群，提出计算弱序中并的递归算法，并证明 Dyer 猜想：并的反演集等于各因子反演集的并集。
An extended generalization of RSK via the combinatorics of type $A$ quiver representations
本文通过揭示 $A$ 型箭图表示的组合学性质，构造了一族由对称群中 Coxeter 元参数化的双射，将整数划分 $λ$ 的任意填充映射到反向平面划分，并在特定参数选择下恢复了经典的 Robinson--Schensted--Knuth 对应。
Canonical reduced words and signed descent length enumeration in Coxeter groups
本文利用 Coxeter 群中典范约化词的下降构造对合，统一推导了对称群及 B、D 型 Coxeter 群中 drops、depth、exc 的带符号枚举公式，并借助 Foata-Zeilberger 双射得到连分数及应用，给出了交错群上 drops 的均值与方差。
Patterns in rectangulations. Part I: $\top$-like patterns, inversion sequence classes $I(010, 101, 120, 201)$ and $I(011, 201)$, and rushed Dyck paths
本文系统定义并研究矩形剖分的模式避免，枚举避免类 $\top$ 模式的弱、强矩形剖分，并建立其与 Catalan 结构、逆序序列类及仓促 Dyck 路径的双射。
Low elements in dominant Shi regions
本文补充了仿射Coxeter群低元素的研究，用Borel子代数的ad-幂零理想解释支配Shi区域最小元素为低元素的原因，并综述了仿射Weyl群支配Shi区域研究中的相关双射。
Let $W$ be an irreducible Coxeter group. We define the Coxeter pop-stack-sorting operator $\mathsf{Pop}:W\to W$ to be the map that fixes the identity element and sends each nonidentity element $w$ to the meet of the elements covered by $w$ in the right weak order. When $W$ is the symmetric group $S_n$, $\mathsf{Pop}$ coincides with the pop-stack-sorting map. Generalizing a theorem about the pop-stack-sorting map due to Ungar, we prove that \[\sup\limits_{w\in W}\left|O_{\mathsf{Pop}}(w)\right|=h,\] where $h$ is the Coxeter number of $W$ (with $h=\infty$ if $W$ is infinite) and $O_f(w)$ denotes…
Iterating the Lehmer code on inversion sequences: Catalan fixed points and finite stabilization
本文研究有限整数序列上的算子$\boldsymbol{\theta}$，刻画其不动点、首次像为不动点的序列，证明其与卡塔兰数、Dyck路径的关联，还证明所有逆序列存在有限稳定化并给出相关性质。
A conjecture on descents, inversions and the weak order
本文提出并证明了关于对称群和超立方群中元素划分与右下降数关系的猜想。
Parabolic Tamari Lattices in Linear Type B
本研究聚焦B型Coxeter群的抛物对齐元，给出其模式避免的组合模型，定义了对应等价关系并证明其为弱序同余，得到了A型抛物Tamari格的B型类似物，填补了相关文献空白。
We show that an element $w$ of a finite Weyl group $W$ is rationally smooth if and only if the hyperplane arrangement $I$ associated to the inversion set of $w$ is inductively free, and the product $(d_1+1) \cdots (d_l+1)$ of the coexponents $d_1,\ldots,d_l$ is equal to the size of the Bruhat interval $[e,w]$, where $e$ is the identity in $W$. As part of the proof, we describe exactly when a rationally smooth element in a finite Weyl group has a chain Billey-Postnikov decomposition. For finite Coxeter groups, we show that chain Billey-Postnikov decompositions are connected with certain modular…
We generalize the Tamari lattice by extending the notions of $231$-avoiding permutations, noncrossing set partitions, and nonnesting set partitions to parabolic quotients of the symmetric group $\mathfrak{S}_{n}$. We show bijectively that these three objects are equinumerous. We show how to extend these constructions to parabolic quotients of any finite Coxeter group. The main ingredient is a certain aligned condition of inversion sets; a concept which can in fact be generalized to any reduced expression of any element in any (not necessarily finite) Coxeter group.
In this paper, we establish a bijection between the infinite reduced words of an affine Weyl group and certain biclosed sets of its positive system and determine all finitely generated biclosed sets in the positive system of an affine Weyl group. Using these results, we show first that the biclosed sets in the standard positive system of rank 3 affine Weyl groups when ordered by inclusion form a complete algebraic ortholattice and secondly that the (generalized) braid graphs of those Coxeter groups are connected, which can be thought of as an infinite version of Tit's solution to the word prob…
Cambrian fans
本文研究了Coxeter群W的c-Cambrian扇形，证明其与广义associahedron的正常扇形在组合上同构，并探讨了c-Cambrian扇形与c-簇的线性同构关系。
Affine Weyl groups and their parabolic quotients are used extensively as indexing sets for objects in combinatorics, representation theory, algebraic geometry, and number theory. Moreover, in the classical Lie types we can conveniently realize the elements of these quotients via intuitive geometric and combinatorial models such as abaci, alcoves, coroot lattice points, core partitions, and bounded partitions. Berg, Jones, and Vazirani described a bijection between n-cores with first part equal to k and (n-1)-cores with first part less than or equal to k, and they interpret this bijection in te…
Middle orders: all distributive lattices between weak and Bruhat
本文研究Coxeter群中位于弱序与Bruhat序之间的分配格（称为中间序），在A型中通过二叉树构造了所有这样的格，并推广到其他Weyl群得到“极小中间序”。
We define and study odd analogues of classical geometric and combinatorial objects associated to permutations, namely odd Schubert varieties, odd diagrams, and odd inversion sets. We show that there is a bijection between odd inversion sets of permutations and acyclic orientations of the Turán graph, that the dimension of the odd Schubert variety associated to a permutation is the odd length of the permutation, and give several necessary conditions for a subset of $[n] \times [n]$ to be the odd diagram of a permutation. We also study the sign-twisted generating function of the odd length over …
We construct an intriguing bijection between $021$-avoiding inversion sequences and $(2413,4213)$-avoiding permutations, which proves a sextuple equidistribution involving double Eulerian statistics. Two interesting applications of this result are also presented. Moreover, this result inspires us to characterize all permutation classes that avoid two patterns of length $4$ whose descent polynomial equals that of separable permutations.
Catalan numbers: from FC elements to classical diagram algebras
本文从 $A_n$ 型 Coxeter 群的全交换元出发，利用典范形式重新推导卡塔兰数、Narayana 数与卡塔兰三角，并构造其与 $n+1$ 弦非交叉图之间保持 Temperley-Lieb 代数拼接乘法的唯一双射及双向算法。
Let $W$ be a Coxeter group, and for $u,v\in W$, let $R_{u,v}(q)$ be the Kazhdan-Lusztig $R$-polynomial indexed by $u$ and $v$. In this paper, we present a combinatorial proof of the inversion formula on $R$-polynomials due to Kazhdan and Lusztig. This problem was raised by Brenti. Based on Dyer's combinatorial interpretation of the $R$-polynomials in terms of increasing Bruhat paths, we reformulate the inversion formula in terms of $V$-paths. By a $V$-path from $u$ to $v$ with bottom $w$ we mean a pair $(Δ_1,Δ_2)$ of Bruhat paths such that $Δ_1$ is a decreasing path from $u$ to $w$ and $Δ_2$ i…
An extended generalization of RSK correspondence via $A$ type quiver representations
本文通过$A$型箭图表示，对任意Coxeter元素构造了从填充到反向平面划分的双射，统一并推广了RSK对应的两种先前推广，发展了箭图表示的组合推论。
Joint extremes of inversions and descents of random permutations
本文研究了随机排列的逆序数与下降数的联合分布，利用Hájek投影和高斯近似证明其位于具有独立边缘的二维Gumbel分布最大吸引域内，并在有限Coxeter群框架下推导出中心极限定理。
Bounded joins of biclosed sets
本文对所有有限Coxeter群证明了Dyer关于正根有界闭集并的猜想，并推广到2-闭集，通过闭包准则和根选择论证刻画了反转集与可达元素。
Extreme Values of Permutation Statistics
本文研究有限Coxeter群中Mahonian和Eulerian分布的极值，通过构造三角阵列并运用大偏差理论证明其Gumbel吸引性，揭示两类分布对样本数$k_n$的不同边界要求。
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