February 29, 2024

Congruence properties of consecutive coefficients in arithmetic progression of Gaussian polynomials

A 2023 result of Eichhorn, Engle, and Kronholm describes an interval of consecutive congruences for p(n, m, N), the function that enumerates the partitions of n into at most m parts, none of which are larger than N, in arithmetic progression.   This function is the partition theoretic interpretation of the coefficient on q^n of the Gaussian polynomial, otherwise known as the q-binomial coefficient.   

In this talk we will considerably expand their result to capture a much larger family of congruences.   We will consider known infinite families of congruences for p(n, m), the function that enumerates the partitions of n into at most m parts, and introduce a related infinite family of congruences for a two-colored partition function.  The result Eichhorn, Engle, and Kronholm becomes a special case of our expanded theorem.  

About the speaker

For her undergraduate degree, Joselyne Aniceto attended the University of Texas at San Antonio, where she studied Math Education. She then enrolled in the master's program at the University of Texas Rio Grande Valley where she started doing research with Dr. Brandt Kronholm on Integer Partitions. She has been invited to present her master's thesis research at the 2022 JMM AMS Special Session on Partition Theory, the AMS Fall 2022 Central Sectional Meeting in the Special Session on the Intersection of Number Theory and Combinatorics, and at the Specialty Seminar in Partition Theory at Michigan Technical University. 

She is currently working on a dissertation project with Dr. Kronholm and a collaborator from the Research Institute for Symbolic Computation in Austria on congruence properties of the coefficients of Gaussian Polynomials. She began this research as a fellow for the Japan Society for the Promotion of Science in the Summer of 2023 at the University of Nagoya.