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Louis W. Kolitsch
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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 k−1∑j=0(−1)j(p(n−j(3j+1)/2)−p(n−j(3j+5)/2−1))=(−1)k−1Mk(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.