论文标题
高阶的彩色motzkin路径
Colored Motzkin Paths of Higher Order
论文作者
论文摘要
订单的Motzkin路径 - $ \ ell $是使用步骤$ u =（1,1）$，$ L =（1,0）$的Motzkin路径的概括，对于每个正整数$ i \ leq \ ell $，$ l =（1,0）$和$ d_i =（1，-i）$。我们通过允许在路径边缘上的各种着色方案来进一步概括 - $ \ ell $ motzkin路径。这些$（\vecα，\vecβ）$ - 彩色motzkin路径可以通过适当的Riordan阵列进行列举，模仿Aigner的技术来处理加泰罗尼亚的类数字。在研究了它们相关的Riordan阵列后，我们在$（\vecα，\vecβ）$彩色motzkin路径和各种研究良好的组合物体之间开发了两种射击。特定的着色方案$（\vecα，\vecβ）$允许我们放置$（\vecα，\vecβ）$ - 彩色motzkin路径进行两次射击，并带有不同的$ k $ -dyck路径的不同亚类Modulo-$ k $，以及大惊小怪的良好路径概括。在$（\vecα，\vecβ）$彩色motzkin路径和$ k $ are树的某些子类之间也开发了一般的两者。
Motzkin paths of order-$\ell$ are a generalization of Motzkin paths that use steps $U=(1,1)$, $L=(1,0)$, and $D_i=(1,-i)$ for every positive integer $i \leq \ell$. We further generalize order-$\ell$ Motzkin paths by allowing for various coloring schemes on the edges of our paths. These $(\vecα,\vecβ)$-colored Motzkin paths may be enumerated via proper Riordan arrays, mimicking the techniques of Aigner in his treatment of Catalan-like numbers. After an investigation of their associated Riordan arrays, we develop bijections between $(\vecα,\vecβ)$-colored Motzkin paths and a variety of well-studied combinatorial objects. Specific coloring schemes $(\vecα,\vecβ)$ allow us to place $(\vecα,\vecβ)$-colored Motzkin paths in bijection with different subclasses of generalized $k$-Dyck paths, including $k$-Dyck paths that remain weakly above horizontal lines $y=-a$, $k$-Dyck paths whose peaks all have the same height modulo-$k$, and Fuss-Catalan generalizations of Fine paths. A general bijection is also developed between $(\vecα,\vecβ)$-colored Motzkin paths and certain subclasses of $k$-ary trees.
