Unveiling the Properties and Relationship of Yellowstone Permutation Sequence

This paper explores a mathematical sequence known as the Yellowstone permutation, introduced by Zumkeller (2004). This sequence, characterized by alternating even and odd integers with prime and composite number patterns, is studied for its unique properties and connections to mathematical structures like Pythagorean triples and quadruples. The research employs descriptive and expository methods to explore the sequence’s nature, establishing it as infinite, containing infinitely many primes, and ensuring that all integers appear at least once. The paper also delves into how the Yellowstone permutation sequence can generate both primitive and non-primitive Pythagorean triples and quadruples. It demonstrates that the expressions derived from terms in this sequence consistently yield these triples and quadruples through a combination of algebraic properties and geometric interpretations. Additionally, the study formulates propositions to clarify the relationships between the terms of the Yellowstone permutation sequence and their behavior, particularly in generating Pythagorean constructs. The findings underscore the sequence's intriguing mathematical characteristics, offering insights into number theory and its potential applications. This work highlights the role of such sequences in exploring deeper mathematical relationships and fostering curiosity in combinatorial number theory.