Rademacher's conjecture and expansions at roots of unity of products generating restricted partitions

On the partial fraction decomposition of the restricted partition generating function

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 upcoming work, giving a resolution to Rademacher's conjecture on the asymptotics of these coefficients.

Asymptotics for the partial fractions of the restricted partition generating function I

The generating function for $p_N(n)$, the number of partitions of $n$ into at most $N$ parts, may be written as a product of $N$ factors. We find the behavior of coefficients in the partial fraction decomposition of this product as $N \to \infty$ by applying the saddle-point method, where the saddle-point we need is associated to a zero of the analytically continued dilogarithm. Our main result disproves a conjecture of Rademacher.

Towards an automation of the circle method

The derivation of the Hardy-Ramanujan-Rademacher formula for the number of partitions of $n$ is reviewed. Next, the steps for finding analogous formulas for certain restricted classes of partitions or overpartiitons is examined, bearing in mind how these calculations can be automated in a CAS. Finally, a number of new formulas of this type which were conjectured with the aid of \emph{Mathematica} are presented along with results of a test for their numerical accuracy.

Some formulae for coefficients in restricted $q$-products

In this paper, we derive some formulae involving coefficients of polynomials which occur quite naturally in the study of restricted partitions. Our method involves a recently discovered sieve technique by Li and Wan (Sci. China. Math. 2010). Based on this method, by considering cyclic groups of different orders we obtain some new results for these coefficients. The general result holds for any group of the form $\mathbb{Z}_{N}$ where $N\in\mathbb{N}$ and expresses certain partial sums of coefficients in terms of expressions involving roots of unity. By specializing $N$ to different values, we …

受限分拆函数最小周期猜想的反例

Counterexamples to the Minimum Period Conjecture for Restricted Partition Functions

针对受限分拆函数系数最小周期的Beck-Sam-Woods猜想，推导系数的精确单位根公式，证明周期的整除上下界均为紧的，并构造出该猜想的一族反例。

整数划分的单纯几何：通过A_{k-1}根系得到一个精确的O(1)公式

The Kaleidoscopic Filter on the Partition Manifold, Part I: The Simplicial Ehrhart, Barycentric Pascal Product, and Continuous Bernoulli Exact Closed Forms for $p(n)$

本文通过将划分多面体嵌入有理多面体理论，推导出整数划分函数p_k(n)的精确O(1)公式，并扩展至无限制划分函数p(n)。