A Generalization of the Euler-Glaisher Partition Bijection
In 1748, Euler published his Introductio in Analysin Inﬁnitorum. Chapter 16 of this work is the ﬁrst systematic study of integer partitions in the mathematical literature. In it, he introduces inﬁnite product generating functions and uses them to derive what is now known as Euler’s partition identity, an English translation of which reads as follows: “The number of different ways a given number can be expressed as the sum of different whole numbers is the same as the number of ways in which the same number can be expressed as the sum of odd numbers, whether the same of different.” In modern terminology, the preceding is rephrased as “the number of partitions of n into distinct parts equals the number of partitions of n into odd parts.” In 1883, J.W.L. Glaisher published the ﬁrst bijective proof of Euler’s partition identity, along with a natural generalization: “the number of partitions of n where no part appears more than m − 1 times equals the number of partitions of n where no part is divisible by m.” By combining a construction of P.A. MacMahon called “partitions of inﬁnity” and knowledge of G.E. Andrews’ “partition ideals of order 1” with Glaisher’s bijective proof of Euler’s identity, we are led to discover a large class of partition identities with straightforward bijective proofs. This is joint work with James Sellers and Gary Mullen of Penn State. All terms will be defined and illustrated with concrete examples, so the required mathematical background will be minimal, and the talk should be accessible to all graduate students.
University of South Carolina Number Theory Seminar
Sills, Andrew V., James Sellers, Gary Mullen.
"A Generalization of the Euler-Glaisher Partition Bijection."
Mathematical Sciences Faculty Presentations.