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arXiv · 2006.16195

Cranks for Ramanujan-type congruences of $k$-colored partitions

Larry Rolen·Zack Tripp·Ian Wagner·Samuel Wilson
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Abstract

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. Here, we provide a framework for discovering and proving such invariants for families of congruences. As a first example, we find a family of crank functions that simultaneously explains most known congruences for colored partition functions. The method used, which utilizes Gritsenko, Skoruppa, and Zagier's powerful recent theory of theta blocks, should also be useful for studying other combinatorial functions.

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Larry Rolen, Zack Tripp, Ian Wagner, Samuel Wilson. 2020-06-29. Cranks for Ramanujan-type congruences of $k$-colored partitions. https://arxiv.org/abs/2006.16195

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