Matricial formulae for partitions

Asymptotic formulae for partition ranks

Using an extension of Wright's version of the circle method, we obtain asymptotic formulae for partition ranks similar to formulae for partition cranks which where conjectured by F. Dyson and recently proved by the first author and K. Bringmann.

Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz)

The purpose of this short article is to announce, and briefly describe, a Maple package, PARTITIONS, that (inter alia) completely automatically discovers, and then proves, explicit expressions (as sums of quasi-polynomials) for pm(n) for any desired m. We do this to demonstrate the power of "rigorous guessing" as facilitated by the quasi-polynomial ansatz.

Exact Formulae for the Fractional Partition Functions

The partition function $p(n)$ has been a testing ground for applications of analytic number theory to combinatorics. In particular, Hardy and Ramanujan invented the "circle method" to estimate the size of $p(n)$, which was later perfected by Rademacher who obtained an exact formula. Recently, Chan and Wang considered the fractional partition functions, defined by $\sum_{n = 0}^\infty p_α(n)x^n := \prod_{k=1}^\infty (1-x^k)^{-α}$. In this paper we use the Rademacher circle method to find an exact formula for $p_α(n)$ and study its implications, including log-concavity and the higher-order gener…

关于划分秩的精确公式

Exact formulae for ranks of partitions

本文通过证明Bringmann公式在特定条件下的精确性，为划分秩的精确公式提供了新方法，并推导出Dyson猜想的新证明途径。

$σ_o\ ext{mex}(n)$ 与 $σ_e\ ext{mex}(n)$ 的算术性质及渐近公式

Arithmetic properties and asymptotic formulae for $σ_o\text{mex}(n)$ and $σ_e\text{mex}(n)$

针对奇、偶最小排除数求和函数，本文研究其模任意 $2$ 的幂的间隙性，证明模 $4$、$8$ 的无限同余式族，并给出 Hardy-Ramanujan 型渐近公式。

Counting birthday collisions using partitions

We use partitions to provide some formulae for counting s-collisions and other events in various forms of the Birthday Problem.