Interpreting the Truncated Pentagonal Number Theorem using Partition Pairs

  • 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 k1j=0(1)j(p(nj(3j+1)/2)p(nj(3j+5)/21))=(1)k1Mk(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 k1 distinct parts all of which are greater than the smallest part in S, and the sum of the parts in SU 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.
Published
2015-06-22
Article Number
P2.55