March 7, 2024

Ramanujan partition congruences and Dyson’s rank

A partition of a positive integer n is a finite nonincreasing sequence of positive integers. For example, the partitions of 4 are: 

• 4

• 3+1

• 2+2

• 2+1+1

• 1+1+1+1         

We may give the sequence of partitions numbers as 1,1,2,3,5,7,11,15,22,30,... Mathematicians are interested in the patterns in this sequence. In 1919, Ramanujan proved the following congruences with q-series. In 1944, Dyson requested proofs of Ramanujan's congruences that "will not appeal to generating functions but will demonstrate by cross-examination of the partitions themselves."  Dyson proposed a statistic called the rank that would do just that.  In this talk, we discuss Ramanujan’s partition congruences and Dyson’s rank.  We extend these ideas to a certain restricted partition function and consider Dyson style witnesses for congruences. 

About the speaker

Jena Gregory earned her bachelor’s degree in 1996 in Mathematics Education.  She graduated from UTRGV in 2021 in Pure Mathematics and is currently in the Ph.D. program.  She taught for 20 years in junior high and high school, helping students of all ages and abilities. She currently works for the Mathematics and Sciences Academy teaching Pre-Calculus and Calculus. She is working on her dissertation with Dr. Brandt Kronholm in partition theory.