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Combinatorial Proofs of Two Euler-Type Identities Due to Andrews

  • Cristina BallantineEmail author
  • Richard Bielak
Article
  • 3 Downloads

Abstract

Let a(n) be the number of partitions of n, such that the set of even parts has exactly one element, b(n) be the difference between the number of parts in all odd partitions of n and the number of parts in all distinct partitions of n, and c(n) be the number of partitions of n in which exactly one part is repeated. Beck conjectured that \(a(n)=b(n)\) and Andrews, using generating functions, proved that \(a(n)=b(n)=c(n)\). We give a combinatorial proof of Andrews’ result. Our proof relies on bijections between a set and a multiset, where the partitions in the multiset are decorated with bit strings. We prove combinatorially Beck’s second conjecture, which was also proved by Andrews using generating functions. Let \(c_1(n)\) be the number of partitions of n, such that there is exactly one part occurring three times, while all other parts occur only once and let \(b_1(n)\) be the difference between the total number of parts in the partitions of n into distinct parts and the total number of different parts in the partitions of n into odd parts. Then, \(c_1(n)=b_1(n)\).

Keywords

Partitions Euler’s identity Bit strings Overpartitions 

Mathematics Subject Classification

05A17 11P81 11P83 

Notes

References

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    Yang, J.Y.X.: Combinatorial proofs and generalizations of conjectures related to Euler’s partition theorem. European J. Combin. 76, 62–72 (2019)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceCollege of the Holy CrossWorcesterUSA

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