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Combinatorial and algebraic properties of Lucas-Analogues
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Resumo(s)
This thesis investigates a generalization of Lucas polynomials through combinatorial and algebraic approaches, including a multivariable extension. Each chapter addresses a unique perspective of these polynomials, building on foundational recurrence relations and exploring their applications in combinatorics and algebra. Chapter 2 develops a combinatorial interpretation of Lucas polynomials, focusing on their representation through tiling problems and other discrete structures. This perspective highlights the connections between recurrence relations and enumeration techniques. Chapter 3 exposes an algebraic framework, examining the properties of Lucas polynomials through transformations and identities. This chapter establishes links with broader algebraic objects and explores how recurrence relations encode these structures. Chapter 4 extends the scope to multivariable Lucas polynomials, presenting a rigorous generalization that integrates combinatorial and algebraic principles. This chapter lays the groundwork for future exploration of multivariable polynomial systems and their applications in higher-dimensional combinatorics. Chapter 5 introduces a recent approach to generating rational functions associated with Lucas polynomials. By transforming previously known recurrence relations, we derive explicit formulas and construct generating functions, offering new insights into these polynomials. This thesis provides a cohesive framework for understanding and extending Lucas polynomials, contributing with ideas for new methodologies and interpretations with relevance to both discrete mathematics and algebraic combinatorics.
Descrição
Tese de Mestrado, Matemática, 2025, Universidade de Lisboa, Faculdade de Ciências
Palavras-chave
Polynomials Lucas-sequence Tiling Lucas Atoms Lucas-Analogue