Polyhedral geometry, supercranks, and combinatorial witnesses of congruences for partitions into three parts

In this paper, we use a branch of polyhedral geometry, Ehrhart theory, to expand our combinatorial understanding of congruences for partition functions. Ehrhart theory allows us to give a new decomposition of partitions, which in turn allows us to define statistics called supercranks that combinatorially witness every instance of divisibility of p(n,3) by any prime m≡−1(mod6), where p(n,3) is the number of partitions of n into three parts. A rearrangement of lattice points allows us to demonstrate with explicit bijections how to divide these sets of partitions into m equinumerous classes. The behavior for primes m′≡1(mod6) is also discussed.