The amplitude of Motzkin paths

限制对合与Motzkin路

Restricted involutions and Motzkin paths

我们展示了如何双射由于Biane之间的对合和标记Motzkin路径产生双射Motzkin路径和两个家庭的限制对合的Motzkin数计数，即对合避免4321和3412。因此，我们得到的Motzkin路径对应的对合，避免4321或3412连同任何模式的长度为3的特征。此外，我们利用所描述的双射来研究限制对合集合的一些显着子集，即无不动点和中心对称限制对合。

非交叉链划分与大（3，2）-Motzkin路

Noncrossing Linked Partitions and Large (3,2)-Motzkin Paths

在自由概率论的某些变换的研究中，出现了非交叉链接划分。我们探讨非交叉链接分区和彩色Motzkin路径之间的连接。（3，2）-Motzkin路径可以被视为彩色Motzkin路径，因为有三种类型的水平阶和两种类型的向下阶。大（3，2）-Motzkin路径被定义为在x轴上仅存在两种类型的水平步长的（3，2）-Motzkin路径。我们建立了[n +1]的非交叉连通划分集与长度为n的大（3，2）-Motzkin路集之间的一一对应。在这种情况下，我们得到了一个简单的解释之间的著名关系的大和小施罗德数。

关于Dyck路径和Motzkin路径的一些结果

Some results on Dyck paths and Motzkin paths

我们在Dyck路集合上引入了一个等价关系，并在Dyck路集合上引入了一些运算.我们确定了这些等价类的基数公式，并使用这些信息来获得一个组合公式的Dyck和Motzkin路径的固定长度的数量。

限制Motzkin置换、Motzkin路、连分数和切比雪夫多项式

Restricted Motzkin permutations, Motzkin paths, continued fractions, and Chebyshev polynomials

我们说一个置换$π$是一个Motzkin置换，如果它避免了132，并且不存在$a<b$使得$π_a<π_b<π_{b+1}$。本文研究了Motzkin置换中几个统计量的分布，包括最长递增和递减序列的长度以及上升和下降的次数。我们还列举了Motzkin置换与额外的限制，并研究了在这一类的置换相当一般的模式的发生的分布。

Some remarks and conjectures about Hankel determinants of polynomials which are related to Motzkin paths

This note collects some results and conjectures for the generating functions of the Hankel determinants of certain polynomials which are related to Motzkin paths.

A bijection between ternary trees and a subclass of Motzkin paths

A bijection between ternary trees with $n$ nodes and a subclass of Motzkin paths of length $3n$ is given. This bijection can then be generalized to $t$-ary trees.

The Pascal Rhombus and the Generalized Grand Motzkin Paths

In the present article, we find a closed expression for the entries of the Pascal rhombus. Moreover, we show a relation between the entries of the Pascal rhombus and a family of generalized grand Motzkin paths.

子空间、子集与Motzkin路径

Subspaces, subsets, and Motzkin paths

该研究定义子空间到Motzkin路径的映射，通过证明路径原像为对称布尔子集不交并，得到子空间格的显式对称布尔分解。

无峰Motzkin路径与LR表之间的一个双射

A bijection between peakless Motzkin paths and LR tableaux

该研究证明了Donnelly等人关于某类Littlewood-Richardson表与长度为n的无峰Motzkin路径数量相等的猜想，并构造了两者间的显式双射。

Motzkin和Schröder路径的逆

Inverses of Motzkin and Schröder Paths

我们建议逆矩阵的三种应用：对于逆Motzkin矩阵，我们看汉克尔行列式，并计算水平带内的路径，对于逆Schröder矩阵，我们看同一带内的路径，但终止于带的顶部。

Combinatorics on bounded free Motzkin paths and its applications

In this paper, we construct a bijection from a set of bounded free Motzkin paths to a set of bounded Motzkin prefixes that induces a bijection from a set of bounded free Dyck paths to a set of bounded Dyck prefixes. We also give bijections between a set of bounded cornerless Motzkin paths and a set of $t$-core partitions, and a set of bounded cornerless symmetric Motzkin paths and a set of self-conjugate $t$-core partitions. As an application, we get explicit formulas for the number of ordinary and self-conjugate $t$-core partitions with a fixed number of corners.

Dyck和Motzkin路径的驼峰

Humps for Dyck and for Motzkin paths

我们计算的总数量的驼峰在戴克和Motzkin路径，我们给标准的杨表格所涉及的数字的解释。然后观察到有趣的现象，驼峰计算改变分区在一个带的分区在一个钩。

S-Motzkin paths with catastrophes and air pockets

So called $S$-Motzkin paths are combined the concepts `catastrophes' and `air pockets. The enumeration is done by properly set up bivariate generating functions which can be extended using the kernel method.

A bijection between two subfamilies of Motzkin paths

Two subfamilies of Motzkin paths, with the same numbers of up, down, horizontal steps were known to be equinumerous with ternary trees and related objects. We construct a bijection between these two families that does not use any auxiliary objects, like ternary trees.

Retakh's Motzkin paths and some combinatorial comments

Dyck paths where peaks are only allowed on level 1 and on even-indexed levels, were introduced by Retakh and analysed by Zeilberger, with assistance from Ekhad. We add some combinatorial comments to the enumeration, which involves Motzkin numbers, in particular, about the average height of such objects.

用于Motzkin路径中的平台的Generating functions

Generating functions for plateaus in Motzkin paths

Motzkin路径中的平台是由三个步骤组成的序列：向上一步，水平一步，然后向下一步。我们发现三种不同形式的二元生成函数的高原Motzkin路径，然后推广到更长的高原。最后，我们描述了进一步的推广：一个连分数的形式，从中可以很容易地得到新的多元生成函数的各种路径统计。生成函数的几个例子给出了使用这种技术。

The $\mathbf{uvu}$-avoiding $(a,b,c)$-Generalized Motzkin paths with vertical steps: bijections and statistic enumerations

A generalized Motzkin path, called G-Motzkin path for short, of length $n$ is a lattice path from $(0, 0)$ to $(n, 0)$ in the first quadrant of the XOY-plane that consists of up steps $\mathbf{u}=(1, 1)$, down steps $\mathbf{d}=(1, -1)$, horizontal steps $\mathbf{h}=(1, 0)$ and vertical steps $\mathbf{v}=(0, -1)$. An $(a,b,c)$-G-Motzkin path is a weighted G-Motzkin path such that the $\mathbf{u}$-steps, $\mathbf{h}$-steps, $\mathbf{v}$-steps and $\mathbf{d}$-steps are weighted respectively by $1, a, b$ and $c$. In this paper, we first give bijections between the set of $\mathbf{uvu}$-avoiding …

标准Young Tableaux和彩色Motzkin路径

Standard Young Tableaux and Colored Motzkin Paths

本文提出了着色Motzkin路的概念，建立了高度有界的$n$ -胞标准Young tableaux（SYT）与长度为$n$的着色Motzkin路之间的一个双射.这一结果不仅给出了标准Young表的格路径解释，而且揭示了至多具有$2d+1$行的SYT集合与至多具有2d行的SYT集合之间的意想不到的内在关系。

Dyck路径与Motzkin路径下面积之和（及其幂次）的显式生成函数

Explicit Generating Functions for the Sum of the Areas Under Dyck and Motzkin Paths (and for Their Powers)

该研究以Motzkin路径为引例，推导广义Dyck路径、Motzkin路径和Dyck路径下面积相关的函数方程，提出处理函数方程的微扰展开算法，可得到任意幂次下两类路径面积幂次之和的显式生成函数。

关于2-距离非交叉划分和加权Motzkin路的注记

A note on 2-distant noncrossing partitions and weighted Motzkin paths

我们证明了Drake和Kim的一个猜想：$\{1,2,...,n\}$的$2$ -远距离非交叉划分的个数等于长度为$n$的Motzkin路的权之和，其中Motzkin路的权是包含斐波那契数的某些分数的乘积.我们给出了猜想的两种证明：一种是利用连分数，另一种是组合证明。

On the duals of the Fibonacci and Catalan-Fibonacci polynomials and Motzkin paths

We use the inversion of coefficient arrays to define dual polynomials to the Fibonacci and Catalan-Fibonacci polynomials, and we explore the properties of these new polynomials sequences. Many of the arrays involved are Riordan arrays. Direct links to the counting of Motzkin paths by different statistics emerge.

加权部分Motzkin路上的矩阵恒等式

Matrix Identities on Weighted Partial Motzkin Paths

本文给出了Catalan数上的矩阵恒等式和Shapiro，Woan和Getu利用Riordan阵列导出的序列$(1, 4, 4^2, 4^3, ...)$的组合解释.通过给出带提升线的加权部分Motzkin路与加权自由Motzkin路之间的一个双射，得到了任意$k \geq 2$的加权Motzkin路个数与序列$(1, k, k^2, k^3, ...)$的一个矩阵恒等式.通过扩展这一论点部分Motzkin路径与多个海拔线，我们给出了一个身份最近获得的卡梅隆和Nkwanta的组合证明。还给出了有色Dyck路上的矩阵恒等式，从而得到序列$(1, t^2+t, (t^2+t)^2, ...)$的矩阵恒等式.

位势多项式与Motzkin路径

Potential Polynomials and Motzkin Paths

长度为$n$的{ \em Motzkin路径}是在平面整数晶格$\mathbb{Z}\times\mathbb{Z}$中从$(0,0)$到$(n,0)$的晶格路径，该晶格路径由水平台阶$(1, 0)$、上台阶$(1,1)$和下台阶$(1,-1)$组成，其从不在x轴下方通过。{ \em $u$ -段{ \rm（resp.} Motzkin路径的$h$ -段{ \rm）}}是连续向上步进的最大序列（{ \rm resp.}水平台阶）。本文研究Motzkin路的两种统计量：“$u$ -段数”和“$h$ -段数”.利用拉格朗日反演公式将Motzkin路径数的加权母函数表示为部分Bell多项式或势多项式之和。作为一个应用，还提供了一个研究组成的一般框架。

Automated Counting of Restricted Motzkin Paths

Motzkin paths are simple yet important combinatorial objects. In this paper, we consider families of Motzkin paths with restrictions on peak heights, valley heights, upward-run lengths, downward-run lengths, and flat-run lengths. This paper presents two fully automated methods for enumerating the paths of such families. The first method uses numeric dynamic programming. While this method often times works, it can be slow and may not work for larger problems. The second method uses symbolic dynamic programming to solve such problems. These methods are implemented in the maple packages accompany…

大 Motzkin 路径与 {0,1,2}-树——一个简单双射

Grand Motzkin paths and $\{0,1,2\}$-trees -- a simple bijection

本文构造大 Motzkin 路径与奇数棵 {0,1,2}-树有序列表之间的简单双射，推广经典 Motzkin 路径对应关系，并为已有工作提供替代证明。

任意权重下狭缝中受限Motzkin路径的生成函数

Generating Functions for Restricted Motzkin Paths in a Slit for Arbitrary Weights

该研究推导了受两条平行线限制、步长为+2/+1/-1的离散Motzkin路径的生成函数，并推广至任意步长权重，拓展了单屏障、均匀步长或权重的有向路径相关研究。

Motzkin路径中的驼峰计数

Counting Humps in Motzkin paths

本文研究了Dyck、Motzkin和Schröder路径中的峰数。最近A. Regev注意到，所有$n$阶Dyck路径中的峰的数目是$n$阶超Dyck路径数目的一半。他还计算了一些驼峰Motzkin路径，并发现了类似的关系，并要求双射证明。我们给出了一个双射并证明了这些结果。利用这个双射，我们还给出了一个新的证明：具有$k$峰的$n$阶Dyck路的个数是Narayana数.通过对超Schröder路的双重计数，我们还得到了一个包含二项式系数乘积的恒等式.

无转角、无峰、无谷的Motzkin路径（规则与斜向）及其在条形图中的应用

Cornerless, peakless, valleyless Motzkin paths (regular and skew) and applications to bargraphs

本文研究无转角Motzkin路径（含斜向变体），通过生成函数和核方法计数UD与DU出现次数，并推广Deutsch-Elizalde双射以应用于条形图前缀分析。

A bijection between the sets of $(a,b,b^2)$-Generalized Motzkin paths avoiding $\mathbf{uvv}$-patterns and $\mathbf{uvu}$-patterns

A generalized Motzkin path, called G-Motzkin path for short, of length $n$ is a lattice path from $(0, 0)$ to $(n, 0)$ in the first quadrant of the XOY-plane that consists of up steps $\mathbf{u}=(1, 1)$, down steps $\mathbf{d}=(1, -1)$, horizontal steps $\mathbf{h}=(1, 0)$ and vertical steps $\mathbf{v}=(0, -1)$. An $(a,b,c)$-G-Motzkin path is a weighted G-Motzkin path such that the $\mathbf{u}$-steps, $\mathbf{h}$-steps, $\mathbf{v}$-steps and $\mathbf{d}$-steps are weighted respectively by $1, a, b$ and $c$. Let $τ$ be a word on $\{\mathbf{u}, \mathbf{d}, \mathbf{v}, \mathbf{d}\}$, denoted …

Involutions under Bruhat order and labeled Motzkin Paths

In this note, we introduce a statistic on Motzkin paths that describes the rank generating function of Bruhat order for involutions. Our proof relies on a bijection introduced by Philippe Biane from permutations to certain labeled Motzkin paths and a recently introduced interpretation of this rank generating function in terms of visible inversions. By restricting our identity to fixed-point-free (FPF) involutions, we recover an identity due to Louis Billera, Lionel Levine and Karola Mészáros with a previous bijective proof by Matthew Watson. Our work sheds new light on the Ethiopian dinner gam…

从$S_n(132)$中的带帽回避置换到广义Dyck和Motzkin路的组合双射

Combinatorial bijections from hatted avoiding permutations in $S_n(132)$ to generalized Dyck and Motzkin paths

我们引入了一个新的避免排列模式的概念，称为帽子模式，这是一个自然的推广的禁止模式。我们显示的增长率的类的排列，避免了帽子模式相比，禁止模式。证明了在$p$高度上无峰的Dyck路、无$ud... du$的Dyck路和Motzkin路都可以用避免$\s_n(132)$中置换的Hatted模式计数，并给出了显式双射.从而给出了无两个相邻数的$\s_n(132)$中Motzkin路与置换之间的一个新的直接双射.这些排列也表示在Motzkin生成树的基础上枚举组合对象（ECO）的方法。

Some statistics on generalized Motzkin paths with vertical steps

Recently, several authors have considered lattice paths with various steps, including vertical steps permitted. In this paper, we consider a kind of generalized Motzkin paths, called {\it G-Motzkin paths} for short, that is lattice paths from $(0, 0)$ to $(n, 0)$ in the first quadrant of the $XOY$-plane that consist of up steps $\mathbf{u}=(1, 1)$, down steps $\mathbf{d}=(1, -1)$, horizontal steps $\mathbf{h}=(1, 0)$ and vertical steps $\mathbf{v}=(0, -1)$. We mainly count the number of G-Motzkin paths of length $n$ with given number of $\mathbf{z}$-steps for $\mathbf{z}\in \{\mathbf{u}, \math…

On the generating functions of pattern-avoiding Motzkin paths

Using a recursive approach, we show that the generating function for sets of Motzkin paths avoiding a single (not necessarily consecutive) pattern is rational over $x$ and the Catalan generating function $C(x) = \frac{1-\sqrt{1-4x^2}}{2x^2}$, where $x$ keeps track of the length of the path. Moreover, an algorithm is provided for finding the generating function in the more general case of an arbitrary set of patterns. In addition, this algorithm allows us to find a combinatorial specification for pattern-avoiding Motzkin paths, which can be used not only for enumeration, but also for exhaustive…

Partial Skew Motzkin Paths

Motzkin paths consist of up-steps, down-steps, level-steps, and never go below the $x$-axis. They return to the $x$-axis at the end. The concept of skew Dyck path \cite{Deutsch-italy} is transferred to skew Motzkin paths, namely, a left step $(-1,-1)$ is additionally allowed, but the path is not allowed to intersect itself. The enumeration of these combinatorial objects was known \cite{Qing}; here, using the kernel method, we extend the results by allowing them to end at a prescribed level $j$. The approach is completely based on generating functions. Asymptotics of the total number of objects…

Identities involving weighted Catalan, Schroder and Motzkin paths

In this paper, we investigate the weighted Catalan, Motzkin and Schröder numbers together with the corresponding weighted paths. The relation between these numbers is illustrated by three equations, which also lead to some known and new interesting identities. To show these three equations, we provide combinatorial proofs. One byproduct is to find a bijection between two sets of Catalan paths: one consisting of those with $k$ valleys, and the other consisting of $k$ $\mathbf{N}$ steps in even positions.

Partial Motzkin paths with air pockets of the first kind avoiding peaks, valleys or double rises

Motzkin paths with air pockets (MAP) of the first kind are defined as a generalization of Dyck paths with air pockets. They are lattice paths in $\mathbb{N}^2$ starting at the origin made of steps $U=(1,1)$, $D_k=(1,-k)$, $k\geq 1$ and $H=(1,0)$, where two down-steps cannot be consecutive. We enumerate MAP and their prefixes avoiding peaks (resp. valleys, resp. double rise) according to the length, the type of the last step, and the height of its end-point. We express our results using Riordan arrays. Finally, we provide constructive bijections between these paths and restricted Dyck and Motzk…

A partial order on Motzkin paths

The Tamari lattice, defined on Catalan objects such as binary trees and Dyck paths, is a well-studied poset in combinatorics. It is thus natural to try to extend it to other families of lattice paths. In this article, we fathom such a possibility by defining and studying an analogy of the Tamari lattice on Motzkin paths. While our generalization is not a lattice, each of its connected components is isomorphic to an interval in the classical Tamari lattice. With this structural result, we proceed to the enumeration of components and intervals in the poset of Motzkin paths we defined. We also ex…

Motzkin 路径中的驼峰与 $(2,1)$-钩中的标准 Young 表

Humps in Motzkin paths and standard Young tableaux in a $(2,1)$-hook

本文统计给定高度 Motzkin 路径的驼峰与峰，以及前两部分差固定的 $(2,1)$-钩标准 Young 表数量，细化 Regev 的结果并给出新组合证明与递推关系。

具有两个长度为2的禁止连续子词的限高Motzkin路径

Motzkin paths of bounded height with two forbidden contiguous subwords of length two

本文在禁止模式背景下研究限高Motzkin路径，通过建立二元生成函数并利用核方法展开，推导出矩阵方程与连分数以完成路径枚举。

一类与加权部分Motzkin路相关的Riordan阵列的子阵

Minors of a Class of Riordan Arrays Related to Weighted Partial Motzkin Paths

部分Motzkin路径是在$XOY$平面中从$(0, 0)$到$(n, k)$的路径，该路径不低于$X$轴，并且由上台阶$U=(1, 1)$、下台阶$D=(1, -1)$和水平台阶$H=(1, 0)$组成。加权部分Motzkin路径是具有权重分配的部分Motzkin路径，其中所有向上步骤和向下步骤都被加权1，如果水平步骤位于$X$轴上，则赋予它们权重$x$，并且如果它们不位于$X$轴上，则赋予它们权重$y$。用$M_{n,k}(x, y)$表示从$(0, 0)$到$(n, k)$的所有加权部分Motzkin路的权函数，$\mathcal{M}=(M_{n,k}(x,y))_{n\geq k\geq 0}$表示无限下三角矩阵。本文考虑矩阵$\mathcal{M}$的二阶子式和，得到了许多与$\mathcal{M}$有关的有趣的行列式恒等式，并利用加权部分Motzkin路用双射证明了这些恒等式.当权参数$(x, y)$被特殊化时，得到了几个与Catalan数相关的新恒等式.此外，在交错情形下，我们也给出了Catalan数的一些新的显式公式。

通过双色 Motzkin 路径精细枚举两行集值标准表

Refined enumeration of two-rowed set-valued standard tableaux via two-coloured Motzkin paths

本文通过建立两行集值标准表与双色 Motzkin 路径的双射，结合生成函数与拉格朗日反演公式，给出了按各行条目数精细计数的显式公式。

The $(s,s+d,\dots,s+pd)$-core partitions and the rational Motzkin paths

In this paper, we propose an $(s+d,d)$-abacus for $(s,s+d,\dots,s+pd)$-core partitions and establish a bijection between the $(s,s+d,\dots,s+pd)$-core partitions and the rational Motzkin paths of type $(s+d,-d)$. This result not only gives a lattice path interpretation of the $(s,s+d,\dots,s+pd)$-core partitions but also counts them with a closed formula. Also we enumerate $(s,s+1,\dots,s+p)$-core partitions with $k$ corners and self-conjugate $(s,s+1,\dots,s+p)$-core partitions.

加权2-Motzkin路

Weighted 2-Motzkin Paths

本文的动机是最近提出的两个问题Coker的组合身份有关的Narayana多项式和Catalan数。我们发现，陈，Deutsch和Elizalde的双射可以用来提供组合解释的身份的Coker时，它适用于加权平面树。为了介绍我们的组合对应关系，我们提供了一个描述的双射陈，Deutsch和Elizalde在一个稍微不同的方式在一个直接建设的形式，从平面树的2-Motzkin路径没有中间步骤涉及的戴克路径。

有界高度的无峰 Motzkin 路径

Peakless Motzkin paths of bounded height

本文研究有界高度的无峰 Motzkin 路径，利用核方法与生成函数奇异性分析给出带状区域内的计数结果，并推导出平均高度渐近式 $2\cdot 5^{-1/4}\sqrt{πn}$。

Two kinds of partial Motzkin paths with air pockets

Motzkin paths with air pockets (MAP) are defined as a generalization of Dyck paths with air pockets by adding some horizontal steps with certain conditions. In this paper, we introduce two generalizations. The first one consists of lattice paths in $\Bbb{N}^2$ starting at the origin made of steps $U=(1,1)$, $D_k=(1,-k)$, $k\geq 1$ and $H=(1,0)$, where two down steps cannot be consecutive, while the second one are lattice paths in $\Bbb{N}^2$ starting at the origin, made of steps $U$, $D_k$ and $H$, where each step $D_k$ and $H$ is necessarily followed by an up step, except for the last step of…

平面树和2-Motzkin路上的奇偶反转对合

Parity Reversing Involutions on Plane Trees and 2-Motzkin Paths

Eu，Liu和Yeh研究了具有$n$边和偶数或奇数叶的平面树的计数问题，并与Stanley关于着色网的一个恒等式联系起来。这个恒等式也是由博宁、夏皮罗和西蒙在他们对施罗德路径的研究中得到的，最近由科克用拉格朗日反演公式推导出来的。一个等价的分区问题是由Klazar独立研究。我们提出了三个奇偶反转对合，一个是无标号的平面树，另一个是标号的平面树，一个是2-Motzkin路，它们与Dyck路一一对应。

Statistics on bargraphs viewed as cornerless Motzkin paths

A bargraph is a self-avoiding lattice path with steps $U=(0,1)$, $H=(1,0)$ and $D=(0,-1)$ that starts at the origin and ends on the $x$-axis, and stays strictly above the $x$-axis everywhere except at the endpoints. Bargraphs have been studied as a special class of convex polyominoes, and enumerated using the so-called wasp-waist decomposition of Bousquet-Mélou and Rechnitzer. In this paper we note that there is a trivial bijection between bargraphs and Motzkin paths without peaks or valleys. This allows us to use the recursive structure of Motzkin paths to enumerate bargraphs with respect to …

基于Motzkin路径生成的无交叉bifix集

Cross-bifix-free sets via Motzkin paths generation

无交叉双缀集合是这样的词集合，即任何词的前缀都不是任何其他词的后缀。本文介绍了一种构造无互双缀定长q元字集的一般方法。它使我们能够确定一个跨bifix自由的词子集，具有不可扩展的属性。

Raney numbers, threshold sequences and Motzkin-like paths

We provide new interpretations for a subset of Raney numbers, involving threshold sequences and Motzkin-like paths with long up and down steps. Given three integers n, k, l such that n >= 1, k >= 2 and 0 <= l <= k-2, a (k,l)-threshold sequence of length n is any strictly increasing sequence S=(s_1 s_2 ... s_n) of integers such that ki <= s_i <= kn+l. These sequences are in bijection with ordered (l+1)-tuples of k-ary trees. We prove this result and identify the Raney numbers that count the (k,l)-threshold sequences. As a consequence, when k=2 and k=3, we deduce combinatorial identities involvi…

具有两种不同水平步的Motzkin路径——核方法方法

Motzkin paths with two variants of level steps on odd levels -- a kernel method approach

本文研究了在奇数层上允许两种水平步的Motzkin路径的计数问题，采用核方法建立生成函数的解析关系，并推导出满足holonomic递推关系的系数序列。

通过Motzkin路径和四分仪游走对区间闭集进行枚举

Enumeration of interval-closed sets via Motzkin paths and quarter-plane walks

本文通过构造双射分别将两条链之积偏序集的区间闭集映射到双色Motzkin路径，将截断矩形偏序集的区间闭集映射到四分仪游走，从而求得了对应的生成函数与函数方程。

Bijections from Dyck and Motzkin meanders with catastrophes to pattern avoiding Dyck paths

In this note, we present constructive bijections from Dyck and Motzkin meanders with catastrophes to Dyck paths avoiding some patterns. As a byproduct, we deduce correspondences from Dyck and Motzkin excursions to restricted Dyck paths.

Vertically constrained Motzkin-like paths inspired by bobbin lace

Inspired by a new mathematical model for bobbin lace, this paper considers finite lattice paths formed from the set of step vectors $\mathfrak{A}=$$\{\rightarrow,$ $\nearrow,$ $\searrow,$ $\uparrow,$ $\downarrow\}$ with the restriction that vertical steps $(\uparrow, \downarrow)$ cannot be consecutive. The set $\mathfrak{A}$ is the union of the well known Motzkin step vectors $\mathfrak{M}=$$\{\rightarrow,$ $\nearrow,$ $\searrow\}$ with the vertical steps $\{\uparrow, \downarrow\}$. An explicit bijection $ϕ$ between the exhaustive set of vertically constrained paths formed from $\mathfrak{A}$ …

Motzkin 路径、321-避免排列以及行奇偶性相同的标准杨表

Motzkin paths, 321-avoiding permutations, and standard Young tableaux with rows of equal parity

本文引入一类由 Motzkin 数计数的 321-避免排列，建立其与 Motzkin 路径及三行标准杨表的双射，使 Riordan 路径对应 Riordan 排列，并连接至更多 Riordan 数计数族。