论文标题
$ \ mathbf {uvu} $ - 避免使用$（a，b，c）$ - 具有垂直步骤的广义motzkin路径：徒
The $\mathbf{uvu}$-avoiding $(a,b,c)$-Generalized Motzkin paths with vertical steps: bijections and statistic enumerations
论文作者
论文摘要
一条通用的motzkin路径，称为g-motzkin路径，长度为$ n $，是一条晶格路径，从$（0，0）$到$（n，0）$在Xoy平面的第一个象限中，由上UP步骤$ \ MATHBF {U} =（u} =（1，1）$，下降步骤$ \ mathbf {dows $ \ mathbf {D} $，1，d} $ \ mathbf {h} =（1，0）$和垂直步骤$ \ mathbf {v} =（0，-1）$。 a $（a，b，c）$ - g-motzkin路径是一条加权的G-Motzkin路径，因此$ \ MathBf {u} $ - 台阶，$ \ Mathbf {h} $ - steps，$ \ Mathbf {v} $ - 步骤 - 在本文中，我们首先在$ \ mathbf {uvu} $ - 避免$（a，b，b^2）$ - g-motzkin长度$ n $的路径与$（a，b）$ - schröder路径以及一组$（a+b，b，b，b，b）$ 2n $ 2n之间， $ \ {\ MathBf {uvu，uu} \} $ - 避免$（a，b，b^2）$ - g-motzkin长度$ n $的路径和一组$（a+b，ab）$ - 长度$ n $的motzkin路径，$ n $，$ \ \ \ \ \ \ \ \ m iv {\ nathbf {uvu} $（a，b，b^2）$ - g-motzkin长度$ n+1 $的$ g-motzkin路径以$ \ mathbf {h} $开头 - 由$ a $和$ a $和$（a，b）$ - 长度$ 2N+2 $的dyck路径的sep加权。在最后一部分中，我们关注统计的枚举“ $ \ MathBf {z} $ - steps”的统计信息。 $ \ mathbf {uvu} $ - 避免使用G-Motzkin路径。这些计数结果与Riordan阵列有关。
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 $(a,b,b^2)$-G-Motzkin paths of length $n$ and the set of $(a,b)$-Schröder paths as well as the set of $(a+b,b)$-Dyck paths of length $2n$, between the set of $\{\mathbf{uvu, uu}\}$-avoiding $(a,b,b^2)$-G-Motzkin paths of length $n$ and the set of $(a+b,ab)$-Motzkin paths of length $n$, between the set of $\{\mathbf{uvu,uu}\}$-avoiding $(a,b,b^2)$-G-Motzkin paths of length $n+1$ beginning with an $\mathbf{h}$-step weighted by $a$ and the set of $(a,b)$-Dyck paths of length $2n+2$. In the last section, we focus on the enumeration of statistics "number of $\mathbf{z}$-steps" for $\mathbf{z}\in \{\mathbf{u}, \mathbf{h}, \mathbf{v}, \mathbf{d}\}$ and "number of points" at given level in $\mathbf{uvu}$-avoiding G-Motzkin paths. These counting results are linked with Riordan arrays.
