论文标题
带有空气口袋的两种部分Motzkin路径
Two kinds of partial Motzkin paths with air pockets
论文作者
论文摘要
带有空气口袋（MAP）的Motzkin路径定义为通过在某些条件下添加一些水平步骤，将带有空气口袋的Dyck路径的概括。在本文中，我们介绍了两个概括。第一个由$ \ bbb {n}^2 $的晶格路径组成，以步骤$ u =（1,1）$，$ d_k =（1，-k）$，$ k \ geq 1 $和$ h =（1,0）$，在第二步中是连续的，而第二步是$ \ bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bbb。 $ u $，$ d_k $和$ h $，其中每个步骤$ d_k $和$ h $都必须采用UP步骤，除了路径的最后一步。我们根据长度，最后一步的类型及其末端的高度为这些路径提供枚举结果。对于从右到左阅读的这些路径也进行了类似的研究。作为副产品，我们获得了由Motzkin数量计数的新类别路径。最后，我们使用Riordan阵列表达结果。
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 the path. We provide enumerative results for these paths according to the length, the type of the last step, and the height of its end-point. A similar study is made for these paths read from right to left. As a byproduct, we obtain new classes of paths counted by the Motzkin numbers. Finally, we express our results using Riordan arrays.
