Drew Armstrong, Sen-Peng Eu
For each skew shape we define a nonhomogeneous symmetric function, generalizing a construction of Pak and Postnikov. In two special cases, we show that the coefficients of this function when expanded in the complete homogeneous basis are given in terms of the (reduced) type of $k$-divisible noncrossing partitions. Our work extends Haiman's notion of a parking function symmetric function.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We consider a variant of the game of Brussels Sprouts that, like Conway's original version, ends in a predetermined number of moves. We show that the endstates of the game are in natural bijection with noncrossing trees and that the game histories are in natural bijection with both parking functions and factorizations of a cycle of $S_n$.
The Chow Polynomial of the Noncrossing Partition Lattice
该研究证明非交叉分拆格的周多项式等于无约束停车函数的下降生成函数，进而得到停车函数下降统计量的新结果。
Consider the noncrossing set partitions of an $n$-element set which either do not contain the block $\{n-1,n\}$, or which do not contain the singleton block $\{n\}$ whenever $1$ and $n-1$ are in the same block. In this article we study the subposet of the noncrossing partition lattice induced by these elements, and show that it is a supersolvable lattice, and therefore lexicographically shellable. We give a combinatorial model for the NBB bases of this lattice and derive an explicit formula for the value of its Möbius function between least and greatest element. This work is motivated by a rec…
On $q$-Counting of Noncrossing Chains and Parking Functions
该研究从$q$-多项式角度探究有限Coxeter群下带权非交叉划分极大链与A、B型停车函数的关联，构建坏边与未停偏好位车辆的双射，解释超八面体群情形下多项式的$γ$-正性。
There are two natural simplicial complexes associated to the noncrossing partition lattice: the order complex of the full lattice and the order complex of the lattice with its bounding elements removed. The latter is a complex that we call the noncrossing partition link because it is the link of an edge in the former. The first author and his coauthors conjectured that various collections of simplices of the noncrossing partition link (determined by the undesired parking spaces in the corresponding parking functions) form contractible subcomplexes. In this article we prove their conjecture by …
There is a well-known bijection between parking functions of a fixed length and maximal chains of the noncrossing partition lattice which we can use to associate to each set of parking functions a poset whose Hasse diagram is the union of the corresponding maximal chains. We introduce a decomposition of parking functions based on the largest number omitted and prove several theorems about the corresponding posets. In particular, they share properties with the noncrossing partition lattice such as local self-duality, a nice characterization of intervals, a readily computable Möbius function, an…
Parking Spaces for Complex Reflection Groups
本文将非交叉停车空间理论推广到不可约良生成复反射群，构造了组合与代数两种停车空间并证明其等变同构，从而枚举非交叉停车函数并推广到 Fuss 情形。
On the structure of the d-indivisible noncrossing partition posets
本文研究d-不可分非交叉划分偏序集，通过生成函数、Hopf代数对映体、EL-标号及d-停车树等工具，获得新计数结果并揭示其结构性质。
Lascoux series, parking functions and noncrossing partitions
本文针对特定排列得到拉斯科多项式的组合解释，发现其与A_{n-1}型考克斯特群非交叉分拆格序复形的h多项式一致，据此提出任意有限考克斯特群h多项式的算子方法，解决了相关公开问题并得到k可除非交叉分拆h多项式的交错对称分解性质。
In 1980, Edelman defined a poset on objects called the noncrossing 2-partitions. They are closely related with noncrossing partitions and parking functions. To some extent, his definition is a precursor of the parking space theory, in the framework of finite reflection groups. We present some enumerative and topological properties of this poset. In particular, we get a formula counting certain chains, that encompasses formulas for Whitney numbers (of both kinds). We prove shellability of the poset, and compute its homology as a representation of the symmetric group. We moreover link it with tw…
Exploring Parking Functions: Poset and Polytope Perspectives
本文针对经典组合对象停车函数，提出从非交叉划分偏序集和多胞体两种视角研究其结构与性质，为停车函数研究提供新的分析角度。
For coprime positive integers $a<b$, Armstrong, Rhoades, and Williams (2013) defined a set $NC(a,b)$ of rational noncrossing partitions, a subset of the ordinary noncrossing partitions of $\{1, \ldots, b-1\}$. Bodnar and Rhoades (2015) confirmed their conjecture that $NC(a,b)$ is closed under rotation and proved an instance of the cyclic sieving phenomenon for this rotation action. We give a definition of $NC(a,b)$ which works for all coprime $a$ and $b$ and prove closure under rotation and cyclic sieving in this more general setting. We also generalize noncrossing parking functions to all cop…
For a fixed integer $k$, we consider the set of noncrossing partitions, where both the block sizes and the difference between adjacent elements in a block is $1\bmod k$. We show that these $k$-indivisible noncrossing partitions can be recovered in the setting of subgroups of the symmetric group generated by $(k+1)$-cycles, and that the poset of $k$-indivisible noncrossing partitions under refinement order has many beautiful enumerative and structural properties. We encounter $k$-parking functions and some special Cambrian lattices on the way, and show that a special class of lattice paths cons…
Let $a < b$ be coprime positive integers. Armstrong, Rhoades, and Williams defined a set $\mathsf{NC}(a,b)$ of `rational noncrossing partitions', which form a subset of the ordinary noncrossing partitions of $\{1, 2, \dots, b-1\}$. Confirming a conjecture of Armstrong et. al., we prove that $\mathsf{NC}(a,b)$ is closed under rotation and prove an instance of the cyclic sieving phenomenon for this rotational action. We also define a rational generalization of the $\mathfrak{S}_a$-noncrossing parking functions of Armstrong, Reiner, and Rhoades.
We test the umbral methods introduced by Rota and Taylor within the theory of representation of symmetric group. We define a simple bijection between the set of all parking functions of length $n$ and the set of all noncrossing partitions of $\{1,2,...,n\}$. Then we give an umbral expression of the Frobenius characteristic of the parking function module introduced by Haiman that allows an explicit relation between this symmetric function and the volume polynomial of Pitman and Stanley.
Real-rootedness and interlacing for parking functions and Chow polynomials
该研究通过转换泊车函数与斯米尔诺夫字的关系，结合递推关系证明了非交叉分拆格Chow多项式等的实根性，解决了相关猜想并得到多个多项式的实根性结论。
Certain mathematical structures make a habit of reoccuring in the most diverse list of settings. Some obvious examples exhibiting this intrusive type of behavior include the Fibonacci numbers, the Catalan numbers, the quaternions, and the modular group. In this article, the focus is on a lesser known example: the noncrossing partition lattice. The focus of the article is a gentle introduction to the lattice itself in three of its many guises: as a way to encode parking functions, as a key part of the foundations of noncommutative probability, and as a building block for a contractible space ac…
The cluster complex on one hand, parking functions on the other hand, are two combinatorial (po)sets that can be associated to a finite real reflection group. Cluster parking functions are obtained by taking an appropriate fiber product (over noncrossing partitions). There is a natural structure of simplicial complex on these objects, and our main goal is to show that it has the homotopy type of a (pure) wedge of spheres. The unique nonzero homology group (as a representation of the underlying reflection group) is a sign-twisted parking representation, which is the same as Gordon's quotient of…
Partial Resolutions and Noncrossing Combinatorics
本文研究了有限可约群的Hecke代数中心元素，通过部分Springer解析和非交叉集合构造，扩展了相关不变量的计算方法。
Noncrossing Combinatorics, the Full Twist, and Decategorification of Knot Invariants
该研究针对纽结不变量，提出去范畴化的组合技术，关联对偶辫群生成元等与纽结不变量，证明$(a,z=0)$-HOMFLYPT多项式可通过该技术计算，还得到非交叉分划等相关模型。
Noncrossing set partitions, nonnesting set partitions, Dyck paths, and rooted plane trees are four classes of Catalan objects which carry a notion of type. There exists a product formula which enumerates these objects according to type. We define a notion of `connectivity' for these objects and prove an analogous product formula which counts connected objects by type. Our proof of this product formula is combinatorial and bijective. We extend this to a product formula which counts objects with a fixed type and number of connected components. We relate our product formulas to symmetric function…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。