An extensive analysis of the parity of broken 3-diamond partitions

Silviu Radu, James A. Sellers

Research output: Contribution to journalArticle

16 Scopus citations

Abstract

In 2007, Andrews and Paule introduced the family of functions δk(n) which enumerate the number of broken k-diamond partitions for a fixed positive integer k. Since then, numerous mathematicians have considered partitions congruences satisfied by δk(n) for small values of k. In this work, we provide an extensive analysis of the parity of the function δ3(n), including a number of Ramanujan-like congruences modulo 2. This will be accomplished by completely characterizing the values of δ3(8n + r) modulo 2 for r ∈ {1, 2, 3, 4, 5, 7} and any value of n ≥ 0. In contrast, we conjecture that, for any integers 0 ≤ B < A, δ3(8(A n + B)) and δ3(8(A n + B) + 6) is infinitely often even and infinitely often odd. In this sense, we generalize Subbarao's Conjecture for this function δ3. To the best of our knowledge, this is the first generalization of Subbarao's Conjecture in the literature.

Original languageEnglish (US)
Pages (from-to)3703-3716
Number of pages14
JournalJournal of Number Theory
Volume133
Issue number11
DOIs
StatePublished - Jul 30 2013

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

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