My research explores partition theory through combinatorial statistics. I study the partition function p(n,m), which counts partitions of n into parts from the set m, and the congruences it satisfies modulo primes. Building on Kronholm’s 2007 discovery of infinite families of such congruences and the 2022 work of Eichhorn, Kronholm, and Larsen identifying cranks that witness them, I have developed new results in which cranks capture sum/difference congruences. My approach combines combinatorial analysis with techniques from Ehrhart theory, particularly properties of the h*-vector, to prove the existence of these cranks for certain primes and small values of m.

In support of this research, I have written two Python programs. The first program, given any integer m and prime ℓ, implements a theorem of Kronholm, Eichhorn, and Larsen that provides a specific algorithm for finding all possible distinct cranks. Many of these cranks are equivalent, meaning they can be expressed as linear combinations of other cranks.  My program distinguishes between distinct and equivalent cranks, grouping them by equivalence classes and listing the representatives of each class. While Eichhorn, Kronholm, and Larsen determined the number of such cranks, they did not have a complete explicit list. This program is therefore significant because it can generate the full set of cranks and organize them according to their equivalences.

The second program, given any integer m and prime ℓ and any specified crank, computes the Ehrhart numerator and then generates the constituents of the resulting quasipolynomial. This capability allows us to observe the sum and difference congruence in action, enabling the generation of large amounts of data that would be infeasible to produce by hand. By examining these constituents, we can identify where crank classes “fix” and“flip,” providing insight into underlying structural behavior. This program is therefore an essential tool for postulating and proving the resulting conjectures in my research.