**Discrete Mathematics Seminar Series**

**Cristina Ballantine**

**Holy Cross**

**Title: ****Partitions decorated with bit strings**

ABSTRACT: Euler's famous partitions identity states that the number of partitions of *n* into distinct parts equals the number of partitions of *n* into odd parts. What happens if we relax the conditions in this identity? Let *a(n)* be the number of partitions of *n* such that the set of even parts has exactly one element and let and *c(n)* be the number of partitions of *n* in which exactly one part is repeated. Moreover, let *b(n)* be the difference between the number of parts in all odd partitions of *n* and the number of parts in all distinct partitions of *n*. Beck conjectured that *a(n)=b(n)* and Andrews, using generating functions, proved that *a(n)=b(n)=c(n)*. Using partitions decorated with bit strings, we give a combinatorial proof of Andrews' result. If time permits, I will introduce some Watson type identities whose combinatorial proofs also use partitions decorated with bit strings.