給定一個 $[m]$ 的子集合 $V$，我們可以定義 peak 的高度在模 $m$ 下不落在 $V$ 裡面的 Dyck path，將長度為 $r$ 的路徑數定義為數列的第 $r$ 項。對任意的正整數 $n$，我們要計算此數列的 $n imes n$ 的 Hankel 行列式，而 Hankel 行列式的值也構成一個新的數列 $H(D^{(m,V)})$。本文的主要結果就是在特定的 $m$ 和 $V$ 之下，Hankel 行列式的值可以完全預測，並且在某些條件下會是週期數列，而且可以預測週期。也可以從可循環的條件構造新的循環條件。更詳細的說，我們的主要定理如下：

我們的第一個主要定理(Theorem 2.3.4.)是當 $m$ 是偶數，且 $V$ 符合條件時，我們可以完全知道 $H(D^{(m,V)})$ 的每一項的值。

第二個主要定理(Theorem 3.3.1.)刻劃了一類集合 $V$，使得當 $m$ 符合特定條件時，$H(D^{(m,V)})$ 會循環，且可以逐項算出其值。

第三個主要定理(Theorem 3.3.3)我們可由上述刻劃的集合，構造出新的集合，使得當 $m$ 符合特定條件時，$H(D^{(m,V)})$ 也會循環。

我們的主要研究方法為分析路徑生成函數 $D^{(m,V)}$，求出 Hankel 行列式的遞迴。
本文相當於得到 Hankel 行列式循環的充分條件，但並非是必要條件。對於我們無法解決的情況，我們也在最後進行討論，並提出一些觀察到的結果與猜想。

For $V subset [m]$, we define a sequence ${d_r^{(m,V)}}$ such that $d_r^{(m,V)}$ is the number of Dyck paths of length $r$ whose peaks avoiding $V$ modulo $m$ and we note that the Hankel determinants of ${d_r^{(m,V)}}$ forms another sequence $H(D^{(m,V)})$. The main results of this thesis are the following:

(1) If $m$ is even, we can compute $H(D^{(m,V)})$ for all $V subset {kin[m] mbox{ } | mbox{ } kmbox{ is even}}$.

(2) Under a suitable assumption on $V$, if $m geq max V$, then $H(D^{(m,V)})$ is periodic and its period can be computed directly.

(3) We provide a method to construct avoiding sets $V$ satisfying the assumption in (2) from a given set $V$ which already satisfies the assumption in (2).

Our approach is analysing the generating functions of $D^{(m,V)}$ to find a recurrence relation for $H(D^{(m,V)})$. For summary, we find sufficient conditions for $H(D^{(m,V)})$ being periodic; however they are not necessary. For those cases still unsolved, we provide partial results from observations and make a conjecture.

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