Lattice Paths and the Triangulations of an n+3-gon
Abstract (300 words maximum)
The Delannoy numbers are a doubly-recursive sequence that enumerates many different mathematical objects. One of these objects is the number of 'Queen's Walks', the number of ways the Queen chess piece moves around the board. We study a generalized version of these numbers, which enumerates lattice paths similar to these chess moves. The Generalized Delannoy Numbers also enumerate certain regions found in a polygon with non-intersecting diagonals. Our methodology involves combinatorial enumeration and bijective analysis to compare these lattice paths with regions formed by non-intersecting diagonals in polygons. We investigated by visually constructing lattice paths and n+3-gons. We used exploratory combinatorial reasoning, mainly through diagram representations, relabeling, and structural pattern recognition to be able to compare these paths with unique regions from the n+3-gons. Our work denotes progress in finding a direct relation between lattice paths and these regions. Through this process, we made progress in identifying a bijection between the Generalized Delannoy lattice paths and the triangular regions of an n+3-gon. Each unique lattice path was able to directly correlate to one unique region of an n+3-gon for each set of polygons where n≥2. These findings can help us better understand how recursive sequences can be scaled and the implications of doing so.
Use of AI Disclaimer
no
Academic department under which the project should be listed
CSM – Mathematics
Primary Investigator (PI) Name
William Griffiths
Lattice Paths and the Triangulations of an n+3-gon
The Delannoy numbers are a doubly-recursive sequence that enumerates many different mathematical objects. One of these objects is the number of 'Queen's Walks', the number of ways the Queen chess piece moves around the board. We study a generalized version of these numbers, which enumerates lattice paths similar to these chess moves. The Generalized Delannoy Numbers also enumerate certain regions found in a polygon with non-intersecting diagonals. Our methodology involves combinatorial enumeration and bijective analysis to compare these lattice paths with regions formed by non-intersecting diagonals in polygons. We investigated by visually constructing lattice paths and n+3-gons. We used exploratory combinatorial reasoning, mainly through diagram representations, relabeling, and structural pattern recognition to be able to compare these paths with unique regions from the n+3-gons. Our work denotes progress in finding a direct relation between lattice paths and these regions. Through this process, we made progress in identifying a bijection between the Generalized Delannoy lattice paths and the triangular regions of an n+3-gon. Each unique lattice path was able to directly correlate to one unique region of an n+3-gon for each set of polygons where n≥2. These findings can help us better understand how recursive sequences can be scaled and the implications of doing so.