Analytic and combinatorial explorations of partitions associated with the Rogers-Ramanujan identities and partitions with initial repetitions

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in ful lment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2016.In this thesis, various partition functions with respect to Rogers-Ramanujan identities and George Andrews' partitions with initial repetitions are studied. Agarwal and Goyal gave a three-way partition theoretic interpretation of the Rogers- Ramanujan identities. We generalise their result and establish certain connections with some work of Connor. Further combinatorial consequences and related partition identities are presented. Furthermore, we re ne one of the theorems of George Andrews on partitions with initial repetitions. In the same pursuit, we construct a non-diagram version of the Keith's bijection that not only proves the theorem, but also provides a clear proof of the re nement. Various directions in the spirit of partitions with initial repetitions are discussed and results enumerated. In one case, an identity of the Euler-Pentagonal type is presented and its analytic proof given.M T 201

Combinatorial Properties of Rogers-Ramanujan-Type Identities Arising from Hall-Littlewood Polynomials

Here we consider the $q$-series coming from the Hall-Littlewood polynomials, \begin{equation*} R_\nu(a,b;q)=\sum_{\substack{\lambda \\[1pt] \lambda_1\leq a}} q^{c|\lambda|} P_{2\lambda}\big(1,q,q^2,\dots;q^{2b+d}\big). \end{equation*} These series were defined by Griffin, Ono, and Warnaar in their work on the framework of the Rogers-Ramanujan identities. We devise a recursive method for computing the coefficients of these series when they arise within the Rogers-Ramanujan framework. Furthermore, we study the congruence properties of certain quotients and products of these series, generalizing the famous Ramanujan congruence \begin{equation*} p(5n+4)\equiv0\pmod{5}. \end{equation*}Comment: 16 pages v2: Minor changes included, to appear in Annals of Combinatoric

Overpartitions, lattice paths and Rogers-Ramanujan identities

We extend partition-theoretic work of Andrews, Bressoud, and Burge to overpartitions, defining the notions of successive ranks, generalized Durfee squares, and generalized lattice paths, and then relating these to overpartitions defined by multiplicity conditions on the parts. This leads to many new partition and overpartition identities, and provides a unification of a number of well-known identities of the Rogers-Ramanujan type. Among these are Gordon's generalization of the Rogers-Ramanujan identities, Andrews' generalization of the G\"ollnitz-Gordon identities, and Lovejoy's ``Gordon's theorems for overpartitions.

Bilateral identities of the Rogers-Ramanujan type

We derive by analytic means a number of bilateral identities of the Rogers-Ramanujan type. Our results include bilateral extensions of the Rogers-Ramanujan and the G\"ollnitz-Gordon identities, and of related identities by Ramanujan, Jackson, and Slater. We give corresponding results for multiseries including multilateral extensions of the Andrews-Gordon identities, of Bressoud's even modulus identities, and other identities. The here revealed closed form bilateral and multilateral summations appear to be the very first of their kind. Given that the classical Rogers-Ramanujan identities have well-established connections to various areas in mathematics and in physics, it is natural to expect that the new bilateral and multilateral identities can be similarly connected to those areas. This is supported by concrete combinatorial interpretations for a collection of four bilateral companions to the classical Rogers-Ramanujan identities.Comment: 25 page