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Speaker: Selcuk Aygin
Date: Friday, October 11
Time: 12 to12:50 p.m.
Ramanujan's congruences for the partition function, ranks and cranks
Let n be a natural number and p(n) denote the partition function, the number of different ways to write n as a sum of a collection of smaller numbers. Ramanujan has proven the following elementary congruences for the partition function:
p(5n+4) = 0 mod 5
p(7n+5) = 0 mod 7
p(11n+6) = 0 mod 11
In this talk we prove the first two of the above congruences using analytic techniques. We then introduce ranks and cranks which give combinatorial interpretations for these congruences. Finally, we will introduce some open problems in the study of partition functions, ranks and cranks.