论文标题
Ramanujan型的曲柄$ k $颜色的分区
Cranks for Ramanujan-type congruences of $k$-colored partitions
论文作者
论文摘要
戴森（Dyson）通过他的排名功能为Ramanujan的分区配对提供了著名的组合解释$ 5 $和7美元，并假设一个不变的解释Ramanujan的所有一致性Modulo $ 5 $，$ 7 $，以及$ 11 $。 Garvan和Andrews-Garvan随后发现了一个不变的曲柄，实现了Dyson的目标。文献中已经知道了许多分区功能一致性的进一步示例。在本文中，我们提供了一个框架，以发现和证明为一致性和分区功能的家庭而存在的这种不变的存在。作为第一个示例，我们找到了一个曲柄函数系列，同时解释了彩色分区功能最著名的一致性。关键的见解是利用Gritsenko，Skoruppa和Zagier引起的最新强大的Theta块理论。此处使用的方法应该在研究其他组合功能的研究中很有用。
Dyson famously provided combinatorial explanations for Ramanujan's partition congruences modulo $5$ and $7$ via his rank function, and postulated that an invariant explaining all of Ramanujan's congruences modulo $5$, $7$, and $11$ should exist. Garvan and Andrews-Garvan later discovered such an invariant called the crank, fulfilling Dyson's goal. Many further examples of congruences of partition functions are known in the literature. In this paper, we provide a framework for discovering and proving the existence of such invariants for families of congruences and partition functions. As a first example, we find a family of crank functions that simultaneously explains most known congruences for colored partition functions. The key insight is to utilize a powerful recent theory of theta blocks due to Gritsenko, Skoruppa, and Zagier. The method used here should be useful in the study of other combinatorial functions.
