Asymptotic expansions for partitions generated by infinite products.

Recently, Debruyne and Tenenbaum proved asymptotic formulas for the number of partitions with parts in ( ) and good analytic properties of the corresponding zeta function, generalizing work of Meinardus. In this paper, we extend their work to prove asymptotic formulas if is a multiset of integers and the zeta function has multiple poles. In particular, our results imply an asymptotic formula for the number of irreducible representations of degree of .

Asymptotics of commuting -tuples in symmetric groups and log-concavity.

Denote by the number of -tuples of elements in the symmetric group with commuting components, normalized by the order of . In this paper, we prove asymptotic formulas for . In addition, general criteria for log-concavity are shown, which can be applied to among other examples.

Log concavity for unimodal sequences.

In this paper, we prove that the number of unimodal sequences of size is log-concave. These are coefficients of a mixed false modular form and have a Rademacher-type exact formula due to recent work of the second author and Nazaroglu on false theta functions. Log-concavity and higher Turán inequalities have been well-studied for (restricted) partitions and coefficients of weakly holomorphic modular forms, and analytic proofs generally require precise asymptotic series with error term.

Conjectures of Sun About Sums of Polygonal Numbers.

In this paper, we consider representations of positive integers as sums of generalized -gonal numbers, which extend the formula for the number of dots needed to make up a regular -gon. We mainly restrict to the case where the sums contain at most four distinct generalized -gonal numbers, with the second -gonal number repeated twice, the third repeated four times, and the last is repeated eight times. For a number of small choices of , Sun conjectured that every positive integer may be written in this form.

On the modularity of certain functions from the Gromov-Witten theory of elliptic orbifolds.

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 provide modular completions for several such functions which involve more complicated objects than ordinary modular forms. In particular, we give new closed formulae for special indefinite theta functions of type (1,2) in terms of products of mock modular forms.

Overpartitions and class numbers of binary quadratic forms.

We show that the Zagier-Eisenstein series shares its nonholomorphic part with certain weak Maass forms whose holomorphic parts are generating functions for overpartition rank differences. This has a number of consequences, including exact formulas, asymptotics, and congruences for the rank differences as well as q-series identities of the mock theta type.

Lifting cusp forms to Maass forms with an application to partitions.

For 2 < k [abstract: see text] we define lifts of cuspidal Poincaré series in S(k)(Gamma(0)(N)) to weight 2 - k harmonic weak Maass forms. This construction answers a question of Dyson by providing the general framework "explaining" Ramanujan's mock theta functions. As an application, we show that the number of partitions of a positive integer n is the "trace" of singular moduli of a Maass form arising from the lift of a weight 4 cusp form corresponding to a Calabi-Yau threefold.