David Perkinson, Qiaoyu Yang, Kuai Yu
A depth-first search version of Dhar's burning algorithm is used to give a bijection between the parking functions of a graph and labeled spanning trees, relating the degree of the parking function with the number of inversions of the spanning tree. Specializing to the complete graph answers a problem posed by R. Stanley.
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Inversions in parking functions
本文通过对称函数理论与有根标记森林双射获得停车函数逆序的q-指数生成函数，推广至单位区间停车函数，并引入概率方法推导逆序总数等统计量公式。
Kreweras proved that the reversed sum enumerator for parking functions of length $n$ is equal to the inversion enumerator for labeled trees on $n+1$ vertices. Recently, Perkinson, Yang, and Yu gave a bijective proof of this equality that moreover generalizes to graphical parking functions. Using a depth-first search variant of Dhar's burning algorithm they proved that the codegree enumerator for $G$-parking functions equals the $κ$-number enumerator for spanning trees of $G$. The $κ$-number is a kind of generalized tree inversion number originally defined by Gessel. We extend the work of Perki…
We study the enumeration problem for different kind of tree parking functions introduced recently, called tree parking functions, tree parking distributions, prime tree parking functions, and prime tree parking distributions, for rooted labelled trees of important combinatorial tree families including labelled ordered, unordered and binary trees. Using combinatorial decompositions of the underlying structures yields, after solving the resulting equations, implicit characterizations of suitable generating functions of the total number of such tree parking functions for trees of size $n$ and $n$…
Classical parking functions can be defined in terms of drivers with preferred parking spaces searching a linear parking lot for an open parking spot. We may consider this linear parking lot as a collection of $n$ vertices (parking spots) arranged in a directed path. We generalize this notion to allow for more complicated "parking lots" and define parking functions on arbitrary directed graphs. We then consider a relationship proved by Lackner and Panholzer between parking functions on trees and "mapping digraphs" and we show that a similar relationship holds when edge orientations are reversed…
Recent results have placed the classical shuffle conjecture of Haglund et al. in a broader context of an infinite family of conjectures about parking functions in any rectangular lattice. The combinatorial side of the new conjectures has been defined using a complicated generalization of the dinv statistic which is composed of three parts and which is not obviously non-negative. Here we simplify the definition of dinv, prove that it is always non-negative, and give a geometric description of the statistic in the style of the classical case. We go on to show that in the n x (n+1) lattice, parki…
On Weary Drivers, Records of Trees, and Parking Functions
本文基于根树的记录概念，重新定义停车函数，并建立根树与停车函数之间的记录保持双射，同时证明两者统计量的等分布性，并探讨了若干计数问题。
We prove a combinatorial formula for Macdonald cumulants which generalizes the celebrated formula of Haglund for Macdonald polynomials. We provide several applications of our formula. Firstly, it gives a new, constructive proof of a strong factorization property of Macdonald polynomials proven recently by the author of this paper. Moreover it proves that Macdonald cumulants are $q,t$--positive in the monomial and in the fundamental quasisymmetric bases. Furthermore, we use our formula to prove the recent higher-order Macdonald positivity conjecture for the coefficients of the Schur polynomials…
For a labeled, rooted tree with edges oriented towards the root, we consider the vertices as parking spots and the edge orientation as a one-way street. Each driver, starting with her preferred parking spot, searches for and parks in the first unoccupied spot along the directed path to the root. If all $n$ drivers park, the sequence of spot preferences is called a parking function. We consider the sequences, called \emph{prime} parking functions, for which each driver parks and each edge in the tree is traversed by some driver after failing to park at her preferred spot. We prove that the tota…
We apply the concept of parking functions to rooted labelled trees and functional digraphs of mappings (i.e., functions $f : [n] \to [n]$) by considering the nodes as parking spaces and the directed edges as one-way streets: Each driver has a preferred parking space and starting with this node he follows the edges in the graph until he either finds a free parking space or all reachable parking spaces are occupied. If all drivers are successful we speak about a parking function for the tree or mapping. We transfer well-known characterizations of parking functions to trees and mappings. Especial…
Parking functions, classically defined in terms of cars with preferred parking spots on a directed path attempting to park there, arise in many combinatorial situations and have seen various generalizations. In particular, parking functions have been defined for general digraphs, which yields many more enumeration problems. For example, in a directed tree whose edges are orientated away from the root, it is unknown in general how the number of parking functions on it changes once the orientation is reversed, even in the case when the tree is a star. We show that this orientation reversal resul…
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 $S_n$.
We recall the occupancy problem introduced by Konheim & Weiss in 1966 and we consider parking functions as hash maps. Each car $c_i$ prefers parking space $p_i$ (the hash map $c_i \mapsto p_i$ with $c_i$ is a key and $p_i$ an index into an array), if $p_i$ is occupied then $c_i$ the next available parking space (the hash table implementation using an open addressing strategy). This paper considers some others hash table implementations like hash tables with linked lists (with parking functions as hash maps). Using the Species Theory, we enumerate by Lagrange inversion those hash tables structu…
Statistics on $\ell$-interval parking functions
本文研究了 $\ell$-区间停车函数的统计性质，包括逆序数、位移和主指标，给出了闭式公式，并证明了 Foata 双射在特定条件下保持该集合，导致统计量等分布。
Tiered tree, Parking function and Postnikov-Shapiro algebra
本文研究分层树、图论停车函数与 Postnikov-Shapiro 代数，建立平行四边形多联骨牌与图论停车函数的双射，并定义完全分层图上的 $\mathcal{S}_{G}$ 空间及用 Whitney 运算刻画分层图。
A polytope related to empirical distributions, plane trees, parking functions, and the associahedron
我们定义了一个n维多面体Pi_n（x），它依赖于参数x_i>0，它的组合性质与经验分布、平面树、平面剖分、停车函数和伴随面体密切相关。特别地，我们给出了Pi_n（x）的体积以及当x_i为整数时Pi_n（x）中整数点的个数的显式公式.给出了Pi_n（x）的两个多面体分解，一个与偏序集的序锥有关，另一个与伴随面体有关。
Trees, Parking Functions and Factorizations of Full Cycles
本文扩展了停车函数、标号树与全循环分解之间的对应关系，引入了树逆序计数器的二元细化并证明其与分解计数器匹配，同时刻画了使Stanley双射成立的全部全循环，并建立了与Haglund弹跳统计量的联系。
A family of bijections between G-parking functions and spanning trees
对于顶点为{0，1，.}的有向图G，n}，G-停车函数是n元组（b_1，.，b_n）的非负整数，使得对于{1，.，n}，则存在U中的顶点j，对于该顶点j，从j到G-U有多于b_j条边。构造了G-停车函数集P_G与G的根为0的生成树集T_G之间的一族双射映射，从而给出了|P_G| = |T_G|的组合证明.
Given an undirected graph $G=(V,E)$, and a designated vertex $q\in V$, the notion of a $G$-parking function (with respect to $q$) was independently developed and studied by various authors, and has recently gained renewed attention. This notion generalizes the classical notion of a parking function associated with the complete graph. In this work, we study properties of {\em maximum} $G$-parking functions and provide a new bijection between them and the set of spanning trees of $G$ with no broken circuit. As a case study, we specialize some of our results to the graph corresponding to the disc…
Symmetry in Tree Parking Distributions
本文利用首次回归分解研究毛虫树停车分布中幸运车辆数与车位偏好频率的联合对称性，构造并证明 Fuss-Catalan 生成函数的 $q,t$-模拟，并推广到任意 $m$ 个统计量。
We provide operadic interpretations for two Hopf subalgebras of the algebra of parking functions. The Catalan subalgebra is identified with the free duplicial algebra on one generator, and the Schröder subalgebra is interpreted by means of a new operad, which we call triduplicial. The noncommutative Lagrange inversion formula is then interpreted in terms of duplicial structures. The generic solution of the noncommutative inversion problem appears as the formal sum of all parking functions. This suggests that combinatorial generating functions derived by functional inversion should be obtainabl…
We introduce an object called a tree growing sequence (TGS) in an effort to generalize bijective correspondences between $G$-parking functions, spanning trees, and the set of monomials in the Tutte polynomial of a graph $G$. A tree growing sequence determines an algorithm which can be applied to a single function, or to the set $\mathcal{P}_{G,q}$ of $G$-parking functions. When the latter is chosen, the algorithm uses splitting operations - inspired by the recursive defintion of the Tutte polynomial - to iteratively break $\mathcal{P}_{G,q}$ into disjoint subsets. This results in bijective map…
Kirchhoff's matrix-tree theorem states that the number of spanning trees of a graph G is equal to the value of the determinant of the reduced Laplacian of $G$. We outline an efficient bijective proof of this theorem, by studying a canonical finite abelian group attached to $G$ whose order is equal to the value of same matrix determinant. More specifically, we show how one can efficiently compute a bijection between the group elements and the spanning trees of the graph. The main ingredient for computing the bijection is an efficient algorithm for finding the unique $G$-parking function (reduce…
We give a combinatorial formula for LLT cumulants of unicellular unilevelled shapes in terms of parking functions and Cayley trees. Our formula implies previously conjectured Schur positivity.
The m-Tamari lattice of F. Bergeron is an analogue of the clasical Tamari order defined on objects counted by Fuss-Catalan numbers, such as m-Dyck paths or (m+1)-ary trees. On another hand, the Tamari order is related to the product in the Loday-Ronco Hopf algebra of planar binary trees. We introduce new combinatorial Hopf algebras based on (m+1)-ary trees, whose structure is described by the m-Tamari lattices. In the same way as planar binary trees can be interpreted as sylvester classes of permutations, we obtain (m+1)-ary trees as sylvester classes of what we call m-permutations. These obje…
Parking on the Random Recursive Tree
本文研究随机递归树上的停车过程，证明相变位于密度0，并给出正车流量高概率出现的临界窗口及二元到达情形下的密度阈值。
Noncommutative Symmetric Functions and Lagrange Inversion
计算了0-Hecke代数自然作用在停车函数上的非交换Frobenius特征，并作为推论得到了非交换Lagrange反演公式的各种形式.
We construct three new combinatorial Hopf algebras based on the Loday-Ronco operations on planar binary trees. The first and second algebras are defined on planar trees and labeled planar trees extending the Loday-Ronco and Malvenuto-Reutenauer Hopf algebras respectively. We show that the latter is bidendriform which implies that is also free, cofree, and self-dual. The third algebra involves a new visualization of parking functions as decorated binary trees; it is also bidendriform, free, cofree, and self-dual, and therefore abstractly isomorphic to the algebra PQSym of Novelli and Thibon. We…
Generalized parking functions, descent numbers, and chain polytopes of ribbon posets
我们考虑反转枚举数I_n（q），它计数标记树或等价地计数停车函数。该多项式是广义停车函数的自然扩张。将q = -1代入这个广义多项式，产生具有特定下降集的排列数。在经典情形下，这个结果意味着公式I_n（-1）= E_n，交替排列的个数。我们根据对合原理给出了这些公式的一个组合证明。我们还给出了这些身份的带状偏序集的广义链多面体的体积的几何解释。这种多面体的体积由广义停车函数的和给出，这类似于Pitman和Stanley的停车函数多面体的体积的表达式。
In this paper, we investigate a parking process on a uniform random rooted plane tree with $n$ vertices. Every vertex of the tree has a parking space for a single car. Cars arrive at independent uniformly random vertices of the tree. If the parking space at a vertex is unoccupied when a car arrives there, it parks. If not, the car drives towards the root and parks in the first empty space it encounters (if there is one). We are interested in asymptotics of the probability of the event that all cars can park when $\lfloor αn \rfloor$ cars arrive, for $α> 0$. We observe that there is a phase tra…
Interpreting the two variable Distance enumerator of the Shi hyperplane arrangement
给出了在$n$维中Shi超平面排列$\Sh_n$的距离计数器的双变量精化$D_{\Sh_n}(q,t)$的系数的解释.这个两个变量的细化是由Stanley \cite{stan-rota}为一般的$r$ -扩展的施超平面安排定义的。当$r=1$ .我们定义了数$(n+1)^{n-1}$的三个自然三维划分。第一个产生于停车功能的长度$n$，第二个从特殊偏序集上的$n$顶点定义的Athanasiadis和第三个从生成树上的$n+1$顶点。我们把这三种划分分别称为停车划分、树偏序集划分和生成树划分。我们发现，停车分区的部分之一是相同的边缘标记树的数量与标签集$\{1,2,...,n\}$上$n+1$未标记的顶点。我们证明了停车分区majorises树偏序集分区和猜想，生成树分区也majorises树偏序集分区。
Parking on trees with a (random) given degree sequence and the Frozen configuration model
本文通过将给定度序列树上的停车过程与有向配置模型自然耦合，重新推导出临界 Bienaymé–Galton–Watson 树上停车相变的位置。
Parking on supercritical geometric Bienaymé--Galton--Watson trees
本文针对超临界几何Bienaymé--Galton--Watson树上的停车过程，基于到达分布的生成函数，给出了判定过程处于亚临界、临界或超临界状态的准则。
In this paper, we view parking functions viewed as labeled Dyck paths in order to study a notion of pattern avoidance first introduced by Remmel and Qiu. In particular we enumerate the parking functions avoiding any set of two or more patterns of length 3, and we obtain a number of well-known combinatorial sequences as a result. Along the way, we find bijections between specific sets of pattern-avoiding parking functions and a number of combinatorial objects such as partitions of polygons and trees with certain restrictions.
A matching $M$ in a multigraph $G=(V,E)$ is said to be uniquely restricted if $M$ is the only perfect matching in the subgraph of $G$ induced by $V(M)$ (i.e., the set of vertices saturated by $M$). For any fixed vertex $x_0$ in $G$, there is a bijection from the set of spanning trees of $G$ to the set of uniquely restricted matchings of size $|V|-1$ in $S(G)-x_0$, where $S(G)$ is the bipartite graph obtained from $G$ by subdividing each edge in $G$. Thus the notion "uniquely restricted matchings of a bipartite graph $H$ saturating all vertices in a partite set $X$" can be viewed as an extensio…
Consider the vector space $\mathbb{K}\mathcal{P}$ spanned by parking functions. By representing parking functions as labeled digraphs, Hivert, Novelli and Thibon constructed a cocommutative Hopf algebra PQSym$^{*}$ on $\mathbb{K}\mathcal{P}$. The product and coproduct of PQSym$^{*}$ are analogous to the product and coproduct of the Hopf algebra NCSym of symmetric functions in noncommuting variables defined in terms of the power sum basis. In this paper, we view a parking function as a word. We shall construct a Hopf algebra PFSym on $\mathbb{K}\mathcal{P}$ with a formal basis $\{M_a\}$ analogo…
This paper studies a generalization of parking functions named $k$-Naples parking functions, where backward movement is allowed. One consequence of backward movement is that the number of ascending $k$-Naples is not the same as the number of descending $k$-Naples. This paper focuses on generalizing the bijections of ascending parking functions with combinatorial objects enumerated by the Catalan numbers in the setting of both ascending and descending $k$-Naples parking functions. These combinatorial objects include Dyck paths, binary trees, triangulations of polygons, and non-crossing partitio…
We establish a phase transition for the parking process on critical Galton--Watson trees. In this model, a random number of cars with mean $m$ and variance $σ^{2}$ arrive independently on the vertices of a critical Galton--Watson tree with finite variance $Σ^{2}$ conditioned to be large. The cars go down the tree towards the root and try to park on empty vertices as soon as possible. We show a phase transition depending on $$ Θ:= (1-m)^2- Σ^2 (σ^2+m^2-m).$$ Specifically, when $m \leq 1$, if $ Θ>0,$ then all but (possibly) a few cars will manage to park, whereas if $Θ<0$, then a positive fracti…
Let $1\leq r\leq n$ and suppose that, when the Depth-first Search Algorithm is applied to a given rooted labelled tree on $n+1$ vertices, exactly $r$ vertices are visited before backtracking. Let $R$ be the set of trees with this property. We count the number of elements of $R$. For this purpose, we first consider a bijection, due to Parkinson, Yang and Yu, that maps $R$ onto the set of parking function with center (defined by the authors in a previous article) of size $r$. A second bijection maps this set onto the set of parking functions with run $r$, a property that we introduce here. We th…
Spanning trees in complete uniform hypergraphs and a connection to extended r-Shi hyperplane arrangements
对于所有r >= 3的完全r-一致超图，我们给出了一个计算其生成树个数的Cayley型公式.类似于完全图的生成树与Parking函数之间的双射，我们得到了完全（r+1）-一致超图的生成树与r-Parking函数之间的双射。我们观察到一个简单的后果，这种双射的数量方面的地区的扩展施安排。
Let $(A_u : u \in \mathbb{B})$ be i.i.d.~non-negative integers that we interpret as car arrivals on the vertices of the full binary tree $ \mathbb{B}$. Each car tries to park on its arrival node, but if it is already occupied, it drives towards the root and parks on the first available spot. It is known that the parking process on $ \mathbb{B}$ exhibits a phase transition in the sense that either a finite number of cars do not manage to park in expectation (subcritical regime) or all vertices of the tree contain a car and infinitely many cars do not manage to park (supercritical regime). We ch…
A refinement of Cayley's formula for trees
一棵顶点全序的有根树的真顶点是比它所有的真后代少小于的顶点。我们用真顶点的个数对几种有标号的有根树和森林进行了计数。我们的结果都表示为多项式P_n（a，b，c）= c（a+（n-1）b+c）（2a+（n-2）b+c）.（（n-1）a+b+c），其对于a=b=c=1简化为（n+1）^{n-1}。我们对真顶点的研究是由A. Postnikov提出的二叉树的钩长公式（arXiv：math.CO/0507163）也被W. Y. C. Chen和L. L. M. Yang（arXiv：math.CO/0507163）和R. R. X. Du和F. Liu（arXiv：math.CO/0501147）。我们的方法给出了一个新的证明杜和刘的结果，并给出了新的钩长公式。我们还找到了多项式P_n（a，b，c）在停车函数方面的解释：我们根据停放在其首选停车位的汽车数量来计算停车函数。
Enumeration of certain subsets of uprooted trees and spherical parking functions
本文研究从完全二部图K_{n+1}删除特定边得到的图族，枚举了无根树子集并推导出计数公式，进而得到球形停车函数个数。
Consider a uniform rooted Cayley tree $T_{n}$ with $n$ vertices and let $m$ cars arrive sequentially, independently, and uniformly on its vertices. Each car tries to park on its arrival node, and if the spot is already occupied, it drives towards the root of the tree and parks as soon as possible. Lackner & Panholzer (arXiv:1504.04972) established a phase transition for this process when $ m \approx \frac{n}{2}$. In this work, we couple this model with a variant of the classical Erdös-Rényi random graph process. This enables us to describe the phase transition for the size of the components of…
Parking functions and chip-firing on hypergraphs
本文将图上的停车函数与筹码发射概念扩展至超图，定义了超图停车函数，证明其极大序列由无环定向刻画，且可由$q$-根生成树计数，并揭示了其与超图筹码发射超稳定构型的等价关系。
We classify recurrent states of the Abelian sandpile model (ASM) on the complete split graph. There are two distinct cases to be considered that depend upon the location of the sink vertex in the complete split graph. This characterisation of decreasing recurrent states is in terms of Motzkin words and can also be characterised in terms of combinatorial necklaces. We also give a characterisation of the recurrent states in terms of a new type of parking function that we call a tiered parking function. These parking functions are characterised by assigning a tier (or colour) to each of the cars,…
Trees, parking functions, syzygies, and deformations of monomial ideals
对于图G，我们构造了两个代数，它们的维数都等于G的生成树的个数。其中一个代数是多项式环模某个单项理想的商，而另一个代数是多项式环模线性型的某个幂的商。我们描述了一组单项式，形成一个线性基础，在这两个代数。基本元素对应于在阿贝尔沙堆模型中自然出现的G-停车函数。这些理想是一般类的单调单项理想及其变形的实例。我们证明了单调单项理想的希尔伯特级数总是被它的变形的希尔伯特级数有界。然后我们定义了一个更一般的单项式理想与偏序集和构造自由分解这些理想。在某些情况下，这些决议与Scarf决议一致。本文证明了单调单项理想的Hilbert级数的几个公式，并研究了它们何时等于形变的Hilbert级数。在附录中，我们讨论了沙堆模型。
Parking functions are a widely studied class of combinatorial objects, with connections to several branches of mathematics. On the algebraic side, parking functions can be identified with the standard monomials of $M_n$, a certain monomial ideal in the polynomial ring $S = {\mathbb K}[x_1, \dots, x_n]$ where a set of generators are indexed by the nonempty subsets of $[n] = \{1,2,\dots,n\}$. Motivated by constructions from the theory of chip-firing on graphs we study generalizations of parking functions determined by $M^{(k)}_n$, a subideal of $M_n$ obtained by allowing only generators correspo…
New combinatorial perspectives on MVP parking functions and their outcome map
本文从组合视角研究MVP停车函数的结果映射，通过反转图子图改进纤维大小界，并揭示与Motzkin路径及Abelian沙堆模型的联系。
The toric g-vector of nestohedra
研究通过将简单多面体的托里克g向量表示为gamma向量的非负整数线性组合，揭示了associahedron、cyclohedron及permutahedron的托里克g向量与123-avoiding停车函数的上升统计之间的关系。
Skeletal generalizations of Dyck paths, parking functions, and chip-firing games
该研究引入由Catalan数计数的k-骨架路径与对应生成树数的k-骨架停车函数，将其推广到非整数参数的梯形路径族，建立参数变化下的双射，关联芯片射击、组合表示论与热带几何。
The Priority Lattice
本文引入了优先级格，证明其最大链由停车函数标记，主理想的最大链由部分停车函数标记，并计算了其格级数和特征多项式。
We consider so-called simple families of labelled trees, which contain, e.g., ordered, unordered, binary and cyclic labelled trees as special instances, and study the global and local behaviour of the number of inversions. In particular we obtain limiting distribution results for the total number of inversions as well as the number of inversions induced by the node labelled j in a random tree of size n.
Consider a uniform random rooted tree on vertices labelled by $[n] = \{1,2,\ldots,n\}$, with edges directed towards the root. We imagine that each node of the tree has space for a single car to park. A number $m \le n$ of cars arrive one by one, each at a node chosen independently and uniformly at random. If a car arrives at a space which is already occupied, it follows the unique path oriented towards the root until it encounters an empty space, in which case it parks there; if there is no empty space, it leaves the tree. Consider $m =[αn]$ and let $A_{n,α}$ denote the event that all $[αn]$ c…
Suppose that $m$ drivers each choose a preferred parking space in a linear car park with $n$ spots. In order, each driver goes to their chosen spot and parks there if possible, and otherwise takes the next available spot if it exists. If all drivers park successfully, the sequence of choices is called a parking function. Classical parking functions correspond to the case $m=n$; we study here combinatorial and probabilistic aspects of this generalized case. We construct a family of bijections between parking functions $\text{PF}(m, n)$ with $m$ cars and $n$ spots and spanning forests $\mathscr{…
Suppose that $m$ drivers each choose a preferred parking space in a linear car park with $n$ spots. In order, each driver goes to their chosen spot and parks there if possible, and otherwise takes the next available spot if it exists. If all drivers park successfully, the sequence of choices is called a parking function. Classical parking functions correspond to the case $m=n$. We investigate various probabilistic properties of a uniform parking function. Through a combinatorial construction termed a parking function multi-shuffle, we give a formula for the law of multiple coordinates in the g…
Generating trees growing on the left for pattern-avoiding inversion sequences
本文提出向左生长的生成树构造，用于从右向左生成避免模式的逆序列，并据此给出避免201、210及010、102的逆序列生成函数的新计算，证明其代数性。
A combinatorial formula for LLT cumulants of melting lollipops in terms of spanning trees
本文给出熔化棒棒糖图LLT累积量的生成树正组合公式，肯定其正性并独立证明Schur正性，完全图情形下用停车函数表达。
Shelling Coxeter-like Complexes and Sorting on Trees
在他们关于“类考克斯特复形”的研究中，巴布森和赖纳引入了一个单纯复形$Δ_T$，它与$n$节点上的每棵树$T$相关联，推广了棋盘复形和A型考克斯特复形。他们证明了当树有$b$叶子时，$Δ_T$是$(n-b-1)$连接的。我们给出了$Δ_T$的$(n-b)$ -骨架的一个壳，从而证明了这个猜想.在这个过程中，我们引入的概念，弱序和反转功能的标签树$T$这意味着壳的$Δ_T$，我们构造这样的反转功能足够大的一类树推导出上述猜想，并恢复棋盘复合体$M_{m,n}$与$n \ge 2m-1$的壳。我们还证明了存在或不存在的一个固定的树的反转函数管辖的网络与树结构承认贪婪排序算法的逆消除，并提供一个反转函数的树，每个顶点的能力至少在其度减一。
Some enumerative properties of parking functions
本文研究停车函数的枚举性质，得到多组统计量联合分布的新结果，该结果可通过双射关系应用于标号森林，还探讨了相关技术的拓展方向。
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.
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