| Summary: | 碩士 === 國立中興大學 === 電機工程學系所 === 102 === Enumerating the total number of constrained lattice paths is a well- established subject in combinatorics and constitutes a crucial research topic in academia. Although the architecture of lattice paths is easily understood, most proposed theorems have not yet been confirmed. The number of lattice paths is identified to enable all of the obtained series to be used to represent binomial coefficients. This study entailed employing combination and algebraic methods to derive a formula from the description of occurring recursions.
Section 1 introduces the research motivation as well as the basic structure of lattice paths and presents a literature review. In relevant literature, the issue of Catalan and Motzkin numbers has been raised. Various studies have proposed increasing the lattice paths in three types of fixed direction and correcting a simple formula.
Section 2 describes the mathematical problems that often occur in combination: the occurrence of Catalan numbers and the difficulty in proving them based on a combination of mathematical analyses. In addition, various applications for Catalan numbers are proposed; for example, the can be employed to solve the Ballot problem.
Section 3 presents an approach to increasing the fixed direction of lattice paths; in addition to the three types of fixed direction, the problem was divided into four categories, and the use of “bijection” the relationship to the latter two issues in discussions and proof.
Section 4 details an extension of the problem of increasing the fixed direction of lattice paths, increasing the restriction of the ban on moving to where y l l 0, 1, 2 or less, this study analyzed l negative integers. Based on the analysis, this paper proposes a formula to solve the problem and prove the formula.
Finally, Section 5 provides the conclusion and recommendations for future research.
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