A partial theta identity of Ramanujan and its number-theoretic interpretation

We will interpret a partial theta identity in Ramanujan’s Lost Notebook as a weighted partition theorem involving partitions into distinct parts with smallest part odd. A special case of this yields a new result on the parity of the number of parts in such partitions, comparable to Euler’s pentagonal numbers theorem. We will provide simple and novel proofs of the weighted partition theorem and the special case. Our proof leads to a companion to Ramanujan’s partial theta identity which we will explain combinatorially.     

On partitions with initial repetitions

We consider George Andrews’ fundamental theorem on partitions with initial repetitions and obtain some partition identities and parity results. A simplified, diagram-free, version of William Keith’s bijective proof of the theorem is presented. Lastly, we obtain extensions and variations of the theorem using a class of Rogers–Ramanujan-type identities for n-color partitions studied by A.K. Agarwal.     

A combinatorial study and comparison of partial theta identities of Andrews and Ramanujan

We provide a simple proof of a partial theta identity of Andrews and study the underlying combinatorics. This yields a weighted partition theorem involving partitions into distinct parts with smallest part odd which turns out to be a companion to a weighted partition theorem involving the same partitions that we recently deduced from a partial theta identity in Ramanujan’s Lost Notebook. We also establish some new partition identities from certain special cases of Andrews’ partial theta identity.     

Analysis of a generalized Lebesgue identity in Ramanujan’s Lost Notebook

We analyze a two-parameter q-series identity in Ramanujan’s Lost Notebook that generalizes the product part of the fundamental one-parameter Lebesgue identity. From reformulations of this two-parameter identity, we deduce new partition theorems including variants of the Gauss triangular number identity and Euler’s pentagonal number theorem. We discuss connections with a partial theta identity of Ramanujan and with several classical results such as those of Sylvester and Göllnitz–Gordon.     

Preface

We will interpret a partial theta identity in Ramanujan’s Lost Notebook as a weighted partition theorem involving partitions into distinct parts with smallest part odd. A special case of this yields a new result on the parity of the number of parts in such partitions, comparable to Euler’s pentagonal numbers theorem. We will provide simple and novel proofs of the weighted partition theorem and the special case. Our proof leads to a companion to Ramanujan’s partial theta identity which we will explain combinatorially.     

A unification of two refinements of Euler’s partition theorem

We obtain a unification of two refinements of Euler’s partition theorem respectively due to Bessenrodt and Glaisher. A specialization of Bessenrodt’s insertion algorithm for a generalization of the Andrews-Olsson partition identity is used in our combinatorial construction.     

Congruences for broken 3-diamond and 7 dots bracelet partitions

Andrews and Paule introduced broken k-diamond partitions by using MacMahon’s partition analysis. Later, Fu found a generalization which he called k dots bracelet partitions. In this paper, with the aid of Farkas and Kra’s partition theorem and a p-dissection identity of f(?q), we derive many congruences for broken 3-diamond partitions and 7 dots bracelet partitions.     

Com plete p-elli ptic integrals and a com putation formula of $$\ pi _ p$$π p for $$ p=4$$ p=4

We study the q-bracket operator of Bloch and Okounkov, recently examined by Zagier and other authors, when applied to functions defined by two classes of sums over the parts of an integer partition. We derive convolution identities for these functions and link both classes of q-brackets through divisor sums. As a result, we generalize Euler’s classic convolution identity for the partition function and obtain an analogous identity for the totient function. As corollaries, we generalize Stanley’s Theorem on the number of ones in all partitions of n, and provide several new combinatorial results.     

Balanced partitions

A famous theorem of Euler asserts that there are as many partitions of n into distinct parts as there are partitions into odd parts. We begin by establishing a less well-known companion result, which states that both of these quantities are equal to the number of partitions of n into even parts along with exactly one triangular part. We then introduce the characteristic of a partition, which is determined in a simple way by the placement of odd parts within the list of all parts. This leads to a refinement of the aforementioned result in the form of a new type of partition identity involving characteristic, distinct parts, even parts, and triangular numbers. Our primary purpose is to present a bijective proof of the central instance of this new type of identity, which concerns balanced partitions—partitions in which odd parts occupy as many even as odd positions within the list of all parts. The bijection is accomplished by means of a construction that converts balanced partitions of 2n into unrestricted partitions of n via a pairing of the squares in the Young tableau.     

Parity in partition identities

This paper considers a variety of parity questions connected with classical partition identities of Euler, Rogers, Ramanujan and Gordon. We begin by restricting the partitions in the Rogers-Ramanujan-Gordon identities to those wherein even parts appear an even number of times. We then take up questions involving sequences of alternating parity in the parts of partitions. This latter study leads to: (1) a bi-basic q-binomial theorem and q-binomial series, (2) a new interpretation of the Rogers-Ramanujan identities, and (3) a new natural interpretation of the fifth-order mock theta functions f ₀(q) along with a new proof of the Hecke-type series representation.     

The minimal excludant and Schmidt's partition theorem

In this work, we study Schmidt's partition theorem in a combinatorial manner, and find a strong refinement which connects the minimal excludant of ordinary partitions to the length of Schmidt's partitions. As a byproduct, we obtain a bivariate form of an identity recorded in Ramanujan's lost notebook.     

Combinatorial proofs of an identity from Ramanujan’s lost notebook and its variations

We give two combinatorial proofs and partition-theoretic interpretations of an identity from Ramanujan’s lost notebook. We prove a special case of the identity using the involution principle. We then extend this into a direct proof of the full identity using a generalization of the involution principle. We also show that the identity can be rewritten into a modified form that we prove bijectively. This fits the identity into Pak’s duality of partition identities proven using the involution principle and partition identities proven bijectively. The original identity was first proven algebraically by Andrews as a consequence of an identity of Rogers’ and combinatorially by Kim, while the modified form of the identity generalizes an identity recently found by Andrews and Warnaar related to the product of partial theta functions.     

The refined lecture hall theorem via abacus diagrams

Bousquet-Mélou & Eriksson’s lecture hall theorem generalizes Euler’s celebrated distinct-odd partition theorem. We present an elementary and transparent proof of a refined version of the lecture hall theorem using a simple bijection involving abacus diagrams.     

Weighted Rogers–Ramanujan partitions and Dyson crank

In this paper, we refine a weighted partition identity of Alladi. We write formulas for generating functions for the number of partitions grouped with respect to a partition statistic other than the norm. We tie our weighted results as well as the different statistics with the crank of a partition. In particular, we prove that the number of partitions into even number of distinct parts whose odd-indexed parts’ sum is n is equal to the number of partitions of n with non-negative crank.     

Classical large deviation theorems on complete Riemannian manifolds

We generalize classical large deviation theorems to the setting of complete, smooth Riemannian manifolds. We prove the analogue of Mogulskii’s theorem for geodesic random walks via a general approach using viscosity solutions for Hamilton–Jacobi equations. As a corollary, we also obtain the analogue of Cramér’s theorem. The approach also provides a new proof of Schilder’s theorem. Additionally, we provide a proof of Schilder’s theorem by using an embedding into Euclidean space, together with Freidlin–Wentzell theory.     

An elliptic analogue of a theorem of Hecke

We revisit the classical theorem of Euler regarding special values of the Riemann zeta function as well as Hecke’s generalization of this to Dirichlet’s \(L\)-functions and derive an elliptic analogue. We also discuss transcendence questions that arise from this analogue.     

Partitions with Non-Repeating Odd Parts and Combinatorial Identities

Continuing our earlier work on partitions with non-repeating odd parts and q-hypergeometric identities, we now study these partitions combinatorially by representing them in terms of 2-modular Ferrers graphs. This yields certain weighted partition identities with free parameters. By special choices of these parameters, we connect them to the Göllnitz-Gordon partitions, and combinatorially prove a modular identity and some parity results. As a consequence, we derive a shifted partition theorem mod 32 of Andrews. Finally we discuss basis partitions in connection with the 2-modular representation of partitions with non-repeating odd parts, and deduce two new parity results involving partial theta series.     

Euler’s broken lines in systems with Carathéodory conditions

We consider Euler’s broken lines in a system with its right-hand side measurable in time and investigate their convergence to trajectories of the system. Counterexamples are given that show that partitions with a small diameter do not guarantee the proximity to the funnel of trajectories. For any Carathéodory function, it is suggested to equip the set of closed subsets of the time interval with a metric. We prove that, under conditions close to Carathéodory ones, the convergence with respect to the metric guarantees the convergence of Euler’s broken lines to the funnel of solutions of the system. As a consequence, it is shown that if the right-hand side is continuous and the sublinear growth condition is satisfied, then a sufficiently small diameter of the partition guarantees the proximity of Euler’s broken line to the funnel of solutions of the system.     

How to determine a partition up to conjugation using multisets of hook lengths

If two partitions are conjugate, their multisets of hook lengths are the same. Then one may wonder whether the multiset of hook lengths of a partition determines a partition up to conjugation. The answer turns out to be no. However, we may add an extra condition under which a given multiset of hook lengths determines a partition uniquely up to conjugation. Herman-Chung, and later Morotti found such a condition. We give an alternative proof of Morotti’s theorem and generalize it.