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Dyson famously provided combinatorial explanations for Ramanujan's partitioncongruences modulo $5$ and $7$ via his rank function, and postulated that aninvariant explaining all of Ramanujan's congruences modulo $5$, $7$, and $11$should exist. Garvan and Andrews-Garvan later discovered such an invariantcalled the crank, fulfilling Dyson's goal. Many further examples of congruencesof partition functions are known in the literature. In this paper, we provide aframework for discovering and proving the existence of such invariants forfamilies of congruences and partition functions. As a first example, we find afamily of crank functions that simultaneously explains most known congruencesfor colored partition functions. The key insight is to utilize a powerfulrecent theory of theta blocks due to Gritsenko, Skoruppa, and Zagier. Themethod used here should be useful in the study of other combinatorialfunctions.

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