Elementary proofs of infinitely many congruences for k-elongated partition diamonds

Robson da Silva, Michael D. Hirschhorn, James A. Sellers

Research output: Contribution to journalArticlepeer-review


In 2007, Andrews and Paule published the eleventh paper in their series on MacMahon's partition analysis, with a particular focus on broken k-diamond partitions. On the way to broken k-diamond partitions, Andrews and Paule introduced the idea of k-elongated partition diamonds. Recently, Andrews and Paule revisited the topic of k-elongated partition diamonds. Using partition analysis and the Omega operator, they proved that the generating function for the partition numbers dk(n) produced by summing the links of k-elongated plane partition diamonds of length n is given by [Formula presented] for each k≥1. A significant portion of their recent paper involves proving several congruence properties satisfied by d1,d2 and d3, using modular forms as their primary proof tool. In this work, our goal is to extend some of the results proven by Andrews and Paule in their recent paper by proving infinitely many congruence properties satisfied by the functions dk for an infinite set of values of k. The proof techniques employed are all elementary, relying on generating function manipulations and classical q-series results.

Original languageEnglish (US)
Article number113021
JournalDiscrete Mathematics
Issue number11
StatePublished - Nov 2022

Bibliographical note

Funding Information:
The first author was supported by São Paulo Research Foundation (FAPESP) (grant no. 2019/14796-8 ). The authors gratefully acknowledge the referees for their thoughtful and beneficial suggestions.

Publisher Copyright:
© 2022 Elsevier B.V.


  • Congruences
  • Generating functions
  • k-elongated diamonds
  • Partitions


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