Discrete Mathematics Seminar Series
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.