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

作者：Radu Silviu; Sellers James A^*

来源：Journal of Number Theory, 2013, 133(11): 3703-3716.

DOI：10.1016/j.jnt.2013.05.009

• Broken k-diamonds
• Congruences
• Modular forms
• Partitions

In 2007, Andrews and Paule introduced the family of functions Delta(k) (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 Delta(k),(n) for small values of k. In this work, we provide an extensive analysis of the parity of the function Delta(3) (n), including a number of Ramanujan-like congruences modulo 2. This will be accomplished by completely characterizing the values of Delta(3) (8n + r) modulo 2 for r E {1, 2, 3,4,5, 7} and any value of n %26gt;= 0. In contrast, we conjecture that, for any integers 0 %26lt;= B %26lt; A, Delta(3) (8(An + B)) and Delta(3)(8(An + B) + 6) is infinitely often even and infinitely often odd. In this sense, we generalize Subbarao%26apos;s Conjecture for this function Delta(3). To the best of our knowledge, this is the first generalization of Subbarao%26apos;s Conjecture in the literature.

• 出版日期2013-11