Related papers: Restricted Overpartitions and concave compositions…
Noting a curious link between Andrews' even-odd crank and the Stanley rank, we adopt a combinatorial approach building on the map of conjugation and continue the study of integer partitions with parts separated by parity. Our motivation is…
By work of Bringmann, Ono, and Rhoades it is known that the generating function of the $M_2$-rank of partitions without repeated odd parts is the so-called holomorphic part of a certain harmonic Maass form. Here we improve the standing of…
The modularity of the partition generating function has many important consequences, for example asymptotics and congruences for $p(n)$. In a series of papers the author and Ono \cite{BO1,BO2} connected the rank, a partition statistic…
Motivated by examples of symmetrically constrained compositions, super convex partitions, and super convex compositions, we initiate the study of partitions and compositions constrained by graph Laplacian minors. We provide a complete…
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…
In this paper, we study modular transformation properties of a certain class of functions with indefinite quadratic forms.
Recently, Andrews, Dixit and Yee introduced partition functions associated with Ramanujan/Watson third order mock theta functions $\omega(q)$ and $\nu(q)$. In this paper, we find several new exact generating functions for those partition…
Unary theta functions have played a significant role in the theory of holomorphic modular forms and modular $L$-functions. A partial theta functions is defined analogously, but the sum is over part of the integer lattice. Such sums fail to…
Extending Sellers' result, Das et al. recently proved some congruence results for generalized overcubic partitions using theta functions and posed some related conjectures. In this paper, we provide a combinatorial proof of a result in…
Andrews recently introduced k-marked Durfee symbols, which are a generalization of partitions that are connected to moments of Dyson's rank statistic. He used these connections to find identities relating their generating functions as well…
It was recently shown that $q\omega(q)$, where $\omega(q)$ is one of the third order mock theta functions, is the generating function of $p_{\omega}(n)$, the number of partitions of a positive integer $n$ such that all odd parts are less…
False theta functions closely resemble ordinary theta functions, however they do not have the modular transformation properties that theta functions have. In this paper, we find modular completions for false theta functions, which among…
In this paper, we study modularity of several functions which naturally arose in a recent paper of Lau and Zhou on open Gromov-Witten potentials of elliptic orbifolds. They derived a number of examples of indefinite theta functions, and we…
We study certain algebras of theta-like functions on partitions, for which the corresponding generating functions give rise to theta functions, quasi-Jacobi forms, Appell-Lerch sums, and false theta functions.
We provide new formulas for the coefficients in the partial fraction decomposition of the restricted partition generating function. These techniques allow us to partially resolve a recent conjecture of Sills and Zeilberger. We also describe…
We study higher rank Jacobi partial and false theta functions (generalizations of the classical partial and false theta functions) associated to positive definite rational lattices. In particular, we focus our attention on certain Kostant's…
In this note, we study the dynamics and associated zeta functions of conformally compact manifolds with variable negative sectional curvatures. We begin with a discussion of a larger class of manifolds known as convex co-compact manifolds…
We derive asymptotic formulae for the coefficients of bivariate generating functions with algebraic and logarithmic factors. Logarithms appear when encoding cycles of combinatorial objects, and also implicitly when objects can be broken…
This paper establishes various variational properties of parametrized versions of two convexity-preserving constructs that were recently introduced in the literature: the proximal composition of a function and a linear operator, and the…
In this paper we compute asymptotics for the coefficients of an infinite class of overpartition rank generating functions. Using these results, we show that $ \overline{N}(a,c,n), $ the number of overpartitions of $ n $ with rank congruent…