Bijective methods and combinatorial studies of problems in partition theory and related areas | Posted on:2012-03-30 | Degree:Ph.D | Type:Dissertation | University:The Pennsylvania State University | Candidate:Fu, Shishuo | Full Text:PDF | GTID:1460390011959489 | Subject:Applied Mathematics | Abstract/Summary: | | 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. | Keywords/Search Tags: | Partition theory, Combinatorial, Chapter, Bijective, Related, Problem | | Related items |
| |
|