PARTITIONS WITH PARTS OCCURRING AT MOST THRICE

Abstract

We study partitions of n into parts that occur at most thrice, with weights whose definition is motivated by an identity of Jacobi. A combinatorial bijection between odd and even partitions of maximum weight is extended to a bijection of ``potholes'' (partitions supplied with extra structure) which is used to show that, when n is not triangular, the numbers of odd and even partitions of any weight are equal. The situation for triangular numbers is also analyzed, and this provides a new proof of Jacobi's identity. Finally, the numbers of potholes are related to a Jacobi theta function, and several other combinatorial connexions are noted.