
Louis W. Kolitsch

Michael Burnette
Keywords:
Partitions, Euler's pentagonal number theorem, Partition pairs
Abstract
In 2012 Andrews and Merca gave a new expansion for partial sums of Euler's pentagonal number series and expressed \[\sum_{j=0}^{k1}(1)^j(p(nj(3j+1)/2)p(nj(3j+5)/21))=(1)^{k1}M_k(n)\] where $M_k(n)$ is the number of partitions of $n$ where $k$ is the least integer that does not occur as a part and there are more parts greater than $k$ than there are less than $k$. We will show that $M_k(n)=C_k(n)$ where $C_k(n)$ is the number of partition pairs $(S, U)$ where $S$ is a partition with parts greater than $k$, $U$ is a partition with $k1$ distinct parts all of which are greater than the smallest part in $S$, and the sum of the parts in $S \cup U$ is $n$. We use partition pairs to determine what is counted by three similar expressions involving linear combinations of pentagonal numbers. Most of the results will be presented analytically and combinatorially.