On flushed partitions and concave compositions

In this work, we give combinatorial proofs for generating functions of two problems, i.e., flushed partitions and concave compositions of even length. We also give combinatorial interpretation of one problem posed by Sylvester involving flushed partitions and then prove it. For these purposes, we first describe an involution and use it to prove core identities. Using this involution with modifications, we prove several problems of different nature, including Andrews' partition identities involving initial repetitions and partition theoretical interpretations of three mock theta functions of third order $f(q)$, $\phi(q)$ and $\psi(q)$. An identity of Ramanujan is proved combinatorially. Several new identities are also established.Comment: 19 page

Dyson's Rank, overpartitions, and weak Maass forms

In a series of papers the first author and Ono connected the rank, a partition statistic introduced by Dyson, to weak Maass forms, a new class of functions which are related to modular forms. Naturally it is of wide interest to find other explicit examples of Maass forms. Here we construct a new infinite family of such forms, arising from overpartitions. As applications we obtain combinatorial decompositions of Ramanujan-type congruences for overpartitions as well as the modularity of rank differences in certain arithmetic progressions.Comment: 24 pages IMRN, accepted for publicatio

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.

Overpartition pairs

An overpartition pair is a combinatorial object associated with the q-Gauss identity and the 1ψ1 summation. In this paper, we prove identities for certain restricted overpartition pairs using Andrews ’ theory of q-difference equations for well-poised basic hypergeometric series and the theory of Bailey chains

Limit Shape of Minimal Difference Partitions and Fractional Statistics

The class of minimal difference partitionsMDP(q) (with gap q) is defined by the condition that successive parts in an integer partition differ from one another by at least q≥0. In a recent series of papers by A. Comtet and collaborators, the MDP(q) ensemble with uniform measure was interpreted as a combinatorial model for quantum systems with fractional statistics, that is, interpolating between the classical Bose–Einstein (q=0) and Fermi–Dirac (q=1) cases. This was done by formally allowing values q∈(0,1) using an analytic continuation of the limit shape of the corresponding Young diagrams calculated for integer q. To justify this “replica-trick”, we introduce a more general model based on a variable MDP-type condition encoded by an integer sequence q=(qi), whereby the (limiting) gap q is naturally interpreted as the Cesàro mean of q. In this model, we find the family of limit shapes parameterized by q∈[0,∞) confirming the earlier answer, and also obtain the asymptotics of the number of parts

Asymptotics, Equidistribution and Inequalities for Partition Functions

This thesis consists of three research projects on asymptotics, equidistribution properties and inequalities for partition and overpartition functions. We start by proving that the number of partitions into squares with an even number of parts is asymptotically equal to that of partitions into squares with an odd number of parts. We further show that, for $n$ large enough, the two quantities are different, and that which of the two is bigger depends on the parity of $n.$ By doing so, we answer a conjecture formulated by Bringmann and Mahlburg (2012). We continue by placing this problem in a broader context and by proving that the same results are true for partitions into any powers. For this, we invoke an estimate on Gauss sums found by Banks and Shparlinski (2015) using the effective lower bounds on center density from the sphere packing problem established by Cohn and Elkies (2003). Finally, we compute asymptotics for the coefficients of an infinite class of overpartition rank generating functions, and we show that $\overline{N}(a,c,n),$ the number of overpartitions of $n$ with rank congruent to $a$ modulo $c,$ is equidistributed with respect to $0\le a< c,$ as $n\to\infty,$ for any $c\ge2.$ In addition, we prove some inequalities between ranks of overpartitions recently conjectured by Ji, Zhang and Zhao (2018), and Wei and Zhang (2018)

Partition congruences by involutions

We present a general construction of involutions on integer partitions which enable us to prove a number of modulo 2 partition congruences