Discovering the Fascinating Pattern in the Geometric Representation of Non-intersecting Chords on the Circle within the Motzkin Sequence - Ibrahim, Haslinda; Mohd Darus, Maizon; Karim, Sharmila (Society of Visual Informatics, 2025)
This study explores the Motzkin sequence and introduces a new sequence, the Wing sequence, derived from a novel geometric representation of directed paths. The main objective is to investigate patterns in the geometric representation of the Motzkin sequence and analyze these representations to construct the Wing sequence. The Motzkin sequence, which counts the number of ways to draw non-intersecting chords between points on a circle serves as the basis for this study. The geometric representation of these chords, particularly for and , is analyzed to reveal patterns and properties that could be modified to generate a new sequence. Based on the analysis, we tried some iterations for and to develop initial ideas for constructing the Wing sequence. The technique involved modifying the geometric representation of the Motzkin sequence to derive the Wing sequence. This process included transforming the non-intersecting chords into a circular representation into a linear arrangement. We then removed any representations without chords and only considered those with chords. Next, we transformed the chords into directed paths. Since these directed paths only connected two points, we combined them to form directed paths that passed through all points at least once. These results identified a pattern between the first and second points, leading to the Fist-Second-Third-Points (FSTP) technique for constructing the Wing sequence. The main findings include deriving a general formula for the Wing sequence, establishing a recurrence relation, and constructing a generating function. These results highlight the applicability of the geometric representation technique in discovering new sequences and enhancing geometric representation techniques in mathematical research. The techniques developed in this study can be applied to other geometric problems, offering researchers a more intuitive understanding of structures. Further exploration of these results may open up various applications for combinatorial structures, such as network routing issues, project scheduling, and key generation involving data security.