Hansraj Gupta's "A technique in partitions" revisited: congruences, cranks, and polyhedral geometry

We revisit a 1975 result of Gupta on a “new” technique for computing values of [Formula: see text], the function enumerating the partitions of n whose part sizes come from a finite set G. We generalize Gupta’s main theorem and then use this result to establish an infinite family of partition congruences. The generating function techniques employed have a direct link to polyhedral geometry, which we then use to prove the existence of combinatorial witnesses, known as “cranks”, for the partition congruences.