On Combinatorics of q–Series and Overpartitions

Abstract

In this thesis, we study the combinatorial interpretations of various q–series identities listed in Chu–Zhang and Slater’s compendium (W. Chu and W. Zhang, Bilateral bailey lemma and Rogers—Ramanujan identities, Advances in Applied Mathematics, 42(3):358–391, 2009 and L.J. Slater, Further identities of the Rogers–Ramanujan type, The Proceedings of the London Mathematical Society, 2(1):147– 167, 1952). We employ various combinatorial tools such as (n + t)–color overpartitions, three-line arrays, split part (n + t)–color partitions, 2– color F–partitions and lattice paths to establish the combinatorial interpretations of many q–series. Further we provide bijections between mentioned combinatorial tools. We then find the combinatorial interpretations of q–series identities involving double sums series related to moduli 4k and 4k + 2 using (n + t)–color partitions. In addition to above, we use the combinatorial tools of color partitions, split color partitions and signed partitions notion to define signed color partitions. This tool is used to provide the combinatorial interpretations of 100 q–series identities listed in Chu–Zhang and Slater’s compendium. Furthermore, by employing the same tool with attaching weight, we interpret four more q–series identities. Finally, in this thesis, we study the arithmetic properties of some q–series identities. We find several congruences for the coefficients of power of q that are in arithmetic progressions modulo powers of 2 and 3.