Interpreting the Truncated Pentagonal Number Theorem using Partition Pairs

In 2012 Andrews and Merca gave a new expansion for partial sums of Euler’s pentagonal number series and expressed k−1 ∑ j=0 (−1)(p(n− j(3j + 1)/2)− p(n− j(3j + 5)/2− 1)) = (−1)Mk(n) where Mk(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 Mk(n) = Ck(n) where Ck(n) is the number of partition pairs (S,U) where S is a partition with parts greater than k, U is a partition with k − 1 distinct parts all of which are greater than the smallest part in S, and the sum of the parts in S ∪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.