论文标题
基于下台Modulo $ k $的高度枚举广义堤防路径
Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo $k$
论文作者
论文摘要
For fixed non-negative integers $k$, $t$, and $n$, with $t < k$, a $k_t$-Dyck path of length $(k+1)n$ is a lattice path that starts at $(0, 0)$, ends at $((k+1)n, 0)$, stays weakly above the line $y = -t$, and consists of steps from the step-set $\{(1, 1), （1，-k）\} $。我们通过考虑$ i $ modulo $ k $的下台数量来列举$ k_t $ -dyck路径的家庭。给定一个元组$（a_1，a_2，\ ldots，a_k）$，我们找到了一个精确的枚举公式，该公式的$ k_t $ -dyck路径长度为$（k+1）n $（k+1）n $，带有$ a_i $ downsteps，以$ i $ i $ modulo $ k $，$ 1 \ leq iq iq i \ leq iq k $ \ leq k $ \ leq k $。给出的证据是通过徒或生成功能来完成的。
For fixed non-negative integers $k$, $t$, and $n$, with $t < k$, a $k_t$-Dyck path of length $(k+1)n$ is a lattice path that starts at $(0, 0)$, ends at $((k+1)n, 0)$, stays weakly above the line $y = -t$, and consists of steps from the step-set $\{(1, 1), (1, -k)\}$. We enumerate the family of $k_t$-Dyck paths by considering the number of down-steps at a height of $i$ modulo $k$. Given a tuple $(a_1, a_2, \ldots, a_k)$ we find an exact enumeration formula for the number of $k_t$-Dyck paths of length $(k+1)n$ with $a_i$ down-steps at a height of $i$ modulo $k$, $1 \leq i \leq k$. The proofs given are done via bijective means or with generating functions.
