Counting paths in corridors using circular Pascal arrays
Shaun V. Ault, Charles Kicey
TL;DR
This paper introduces a novel approach connecting circular Pascal arrays with lattice path enumeration within corridors, offering new proofs for existing formulas and insights into path counting problems.
Contribution
It establishes a direct relationship between circular Pascal arrays and lattice paths, providing simplified proofs of complex combinatorial formulas.
Findings
Established a link between circular Pascal arrays and corridor lattice paths
Provided new, concise proofs for known lattice path formulas
Enhanced understanding of path counting using array periodicity
Abstract
A circular Pascal array is a periodization of the familiar Pascal's triangle. Using simple operators defined on periodic sequences, we find a direct relationship between the ranges of the circular Pascal arrays and numbers of certain lattice paths within corridors, which are related to Dyck paths. This link provides new, short proofs of some nontrivial formulas found in the lattice-path literature.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Cellular Automata and Applications · Quasicrystal Structures and Properties