Bijective Methods And Combinatorial Studies Of Problems In Partition Theory And Related Areas

This dissertation explores five problems that arise in the course of studying basic hypergeometric series and enumerative combinatorics, partition theory in particular. Chapter 1 gives a quick introduction to each topic and states the main results. Then each problem is discussed separately in full detail in Chapter 2 through Chapter 6. Chapter 2 starts with Bressound's conjecture, which states that two sets of partitions under certain constraints are equinumerous. The validity of the conjecture in the first two cases implies exactly the partition-theoretical interpretation for the Rogers-Ramanujan identities. We give a nearly bijective proof of the conjecture, and we provide examples to demonstrate the bijection as well. Chapter 3 preserves this combinatorial flavor and supplies a purely combinatorial proof of one congruence that was first obtained by Andrews and Paule in one of their series papers on MacMahon's partition analysis. Chapter 4 addresses an enumeration problem from graph theory and completely solves the problem with a closed formula. Chapter 5 introduces a (q,t)-analogue of binomial coefficient that was first studied by Reiner and Stanton. We also settles a conjecture made by them concerning the sign of each term in this (q,t)-binomial coefficient when q <= -2 is a negative integer. Chapter 6 focuses on two lacunary partition functions and we reproves two related identities uniformly using the orthogonality of the Little q-Jacobi Polynomial. We concludes in Chapter 7 by addressing the significance of bijective and combinatorial methods in the study of partition theory and related areas.