A partition of a positive integer n is a non-increasing sequence of positive integers that sum to n. For example the partitions of 4 are 4, 3+1, 2+2, 2+1+1, and 1+1+1+1. When given a table of values of p(n), the number of partitions of n, Ramanujan immediately conjectured that 5|p(5n+4), 7|p(7n+5), and 11|p(11n+6). The partition function satisfies many other congruences and many functions related to p(n) also satisfy linear congruences along arithmetic progressions. However after 100 years and various theory from q-series, modular forms, and combinatorics one finds that not all congruences are created equal. We give a brief overview of these topics and then focus on ranks and cranks.
The idea of ranks and cranks is best illustrated with the original rank of a partition defined by Dyson 60 years ago. The rank of a partition is the largest part minus the number of parts. If one splits up the partitions of 5n+4 according to the value of their rank taken modulo 5, then one has five sets of equal size (and so clearly p(5n+4) is divisible by 5). The difference between a rank and crank is just word play.
We look at a recent example given by congruences for the number of smallest parts in the overpartitions on n. The congruences for this function have two proofs. The first proof is due to Bringmann, Lovejoy, and Osburn using quasi-modular forms in 2010. The second proof is due to Garvan and the author defining a crank on the smallest parts in the overpartitions on n. We focus on the latter, however both proofs are far from trivial.