Partition identities arising from Ramanujan’s formulas for multipliers

We find new partition identities arising from Ramanujan’s formulas of multipliers. Several of the identities are for overpartitions, overpartition pairs, and \(\ell \)-regular partitions.     

Overpartition pairs and two classes of basic hypergeometric series

We study the combinatorics of two classes of basic hypergeometric series. We first show that these series are the generating functions for certain overpartition pairs defined by frequency conditions on the parts. We then show that when specialized these series are also the generating functions for overpartition pairs with bounded successive ranks, overpartition pairs with conditions on their Durfee dissection, as well as certain lattice paths. When further specialized, the series become infinite products, leading to numerous identities for partitions, overpartitions, and overpartition pairs.     

Rank and Conjugation for a Second Frobenius Representation of an Overpartition

The notions of rank and conjugation are developed in the context of a second Frobenius representation of an overpartition. Some identities for overpartitions that are invariant under this conjugation are presented. Jeremy Lovejoy: The author was partially supported by an ACI “Jeunes Chercheurs et Jeunes Chercheuses”.     

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 Göllnitz-Gordon identities, and Lovejoy's “Gordon's theorems for overpartitions.”     

On the number of partitions of n into k different parts

We study the number of partitions of n into k different parts by constructing a generating function. As an application, we will prove mysterious identities involving convolution of divisor functions and a sum over partitions. By using a congruence property of the overpartition function, we investigate values of a certain convolution sum of two divisor functions modulo 8.     

Some properties of k-tuple t-core partitions

The Ramanujan Journal - An overpartition is a partition in which the first occurrence of a number may be overlined. For an overpartition $$\lambda $$ , let $$\ell (\lambda )$$ denote the largest...     

Asymptotics for Bailey-type mock theta functions

The Ramanujan Journal - We compute asymptotic estimates for the Fourier coefficients of two mock theta functions, which come from Bailey pairs derived by Lovejoy and Osburn. To do so, we employ the...     

Rademacher-type formulas for restricted partition and overpartition functions

A collection of Hardy-Ramanujan-Rademacher type formulas for restricted partition and overpartition functions is presented, framed by several biographical anecdotes.     

Some further <Emphasis Type="Italic">q</Emphasis>-series identities related to divisor functions

We give new generalizations of some q-series identities of Dilcher and Prodinger related to divisor functions. Some interesting special cases are also deduced, including an identity related to overpartitions studied by Corteel and Lovejoy.     

Rank and Conjugation for the Frobenius Representation of an Overpartition

We discuss conjugation and Dyson’s rank for overpartitions from the perspective of the Frobenius representation. More specifically, we translate the classical definition of Dyson’s rank to the Frobenius representation of an overpartition and define a new kind of conjugation in terms of this representation. We then use q-series identities to study overpartitions that are self-conjugate with respect to this conjugation. Received June 28, 2004     

On three third order mock theta functions and Hecke-type double sums

We obtain four Hecke-type double sums for three of Ramanujan’s third order mock theta functions. We discuss how these four are related to the new mock theta functions of Andrews’ work on q-orthogonal polynomials and Bringmann, Hikami, and Lovejoy’s work on unified Witten–Reshetikhin–Turaev invariants of certain Seifert manifolds. We then prove identities between these new mock theta functions by first expressing them in terms of the universal mock theta function.     

A Rogers-Ramanujan bijection

In this paper we give a bijection between the partitions of n with parts congruent to 1 or 4 (mod 5) and the partitions of n with parts differing by at least 2. This bijection is obtained by a cut-and-paste procedure which starts with a partition in one class and ends with a partition in the other class. The whole construction is a combination of a bijection discovered quite early by Schur and two bijections of our own. A basic principle concerning pairs of involutions provides the key for connecting all these bijections. It appears that our methods lead to an algorithm for constructing bijections for other identities of Rogers-Ramanujan type such as the Gordon identities.     

Ramanujan type identities and congruences for partition pairs

Using elementary methods, we establish several new Ramanujan type identities and congruences for certain pairs of partition functions.     

Some congruences modulo 5 and 25 for overpartitions

We present two new Ramanujan-type congruences modulo 5 for overpartitions. We also give an affirmative answer to a conjecture of Dou and Lin, which includes four congruences modulo 25 for overpartition.     

Log-concavity of the overpartition function

We prove that the overpartition function \( \overline{p}(n)\) is log-concave for all \( n\ge 2 \). The proof is based on Sills-Rademacher-type series for \( \overline{p}(n)\) and inspired by DeSalvo and Pak’s proof for the partition function.     

A proof of some conjectures of Mao on partition rank inequalities

Based on work of Atkin and Swinnerton-Dyer on partition rank difference functions, and more recent work of Lovejoy and Osburn, Mao has proved several inequalities between partition ranks modulo 10, and additional results modulo 6 and 10 for the \(M_2\) rank of partitions without repeated odd parts. Mao conjectured some additional inequalities. We prove some of Mao’s rank inequality conjectures for both the rank and the \(M_2\) rank modulo 10 using elementary methods.     

Extending Theorems of Göllnitz,A New Family of Partition Identities

A new family of partition identities is given which include as special cases two theorems of Göllnitz. We show, also, a relation between our result and a theorem given by Sylvester.     

Extension of the partition sieve

We demonstrate the correspondence which lies behind certain partition identities used by Andrews in his partition sieve. This leads to an extension of his methods and a generalization of his results.     

A reciprocity relation for WP-Bailey pairs

We derive a new general transformation for WP-Bailey pairs by considering a certain limiting case of a WP-Bailey chain previously found by the authors, and examine several consequences of this new transformation. These consequences include new summation formulae involving WP-Bailey pairs. Other consequences include new proofs of some classical identities due to Jacobi, Ramanujan and others, and indeed extend these identities to identities involving particular specializations of arbitrary WP-Bailey pairs.