论文标题
筹集了$ k $ -dyck的路径
Raised $k$-Dyck paths
论文作者
论文摘要
提出的$ k $ -dyck路径是$ k $ -dyck路径的概括，可能以非零的高度开始和结束。在本文中，我们为所有高度对$ k $ k $ dyck路径的数量从$（0，α）$（\ ell，β）$的所有高度对$α，β\ geq 0 $，所有长度$ \ ell \ geq 0 $和所有$ k \ geq 2 $ heped $ k $ geq 0 $。然后，我们列举列出$ k $ -dyck路径，其固定数量的收益率是地面的，固定的最小高度和固定的最大高度，呈现生成功能（就生成功能而言，$ k $ -catalan数字的生成函数$ C_K（T）$不可易于易处理。将我们的结果擅长于$ k = 2 $或$α<k $，揭示了与高度结合的戴克路径和“负边界负边界的Dyck路径”的相关结果。
Raised $k$-Dyck paths are a generalization of $k$-Dyck paths that may both begin and end at a nonzero height. In this paper, we develop closed formulas for the number of raised $k$-Dyck paths from $(0,α)$ to $(\ell,β)$ for all height pairs $α,β\geq 0$, all lengths $\ell \geq 0$, and all $k \geq 2$. We then enumerate raised $k$-Dyck paths with a fixed number of returns to ground, a fixed minimum height, and a fixed maximum height, presenting generating functions (in terms of the generating functions $C_k(t)$ for the $k$-Catalan numbers) when closed formulas aren't tractable. Specializing our results to $k=2$ or to $α< k$ reveal connections with preexisting results concerning height-bounded Dyck paths and "Dyck paths with a negative boundary", respectively.
