Radu Silviu, Sellers James A
Research Institute for Symbolic Computation (RISC), Johannes Kepler University, A-4040 Linz, Austria.
Department of Mathematics, Penn State University, University Park, PA 16802, USA.
J Number Theory. 2013 Nov;133(11):3703-3716. doi: 10.1016/j.jnt.2013.05.009.
In 2007, Andrews and Paule introduced the family of functions [Formula: see text] which enumerate the number of broken -diamond partitions for a fixed positive integer . Since then, numerous mathematicians have considered partitions congruences satisfied by [Formula: see text] for small values of . In this work, we provide an extensive analysis of the parity of the function [Formula: see text], including a number of Ramanujan-like congruences modulo 2. This will be accomplished by characterizing the values of [Formula: see text] modulo 2 for [Formula: see text] and any value of [Formula: see text]. In contrast, we conjecture that, for any integers [Formula: see text], [Formula: see text] and [Formula: see text] is infinitely often even and infinitely often odd. In this sense, we generalize Subbarao's Conjecture for this function [Formula: see text]. To the best of our knowledge, this is the first generalization of Subbarao's Conjecture in the literature.
2007年，安德鲁斯（Andrews）和保勒（Paule）引入了函数族[公式：见原文]，该函数族用于枚举固定正整数的破菱形划分的数量。从那时起，许多数学家研究了对于较小的[公式：见原文]值，[公式：见原文]所满足的划分同余关系。在这项工作中，我们对函数[公式：见原文]的奇偶性进行了广泛分析，包括一些模2的类似拉马努金的同余关系。这将通过刻画[公式：见原文]对于[公式：见原文]以及[公式：见原文]的任意值模2的值来实现。相比之下，我们推测，对于任意整数[公式：见原文]、[公式：见原文]，[公式：见原文]无限次为偶数且无限次为奇数。从这个意义上说，我们对这个函数[公式：见原文]推广了苏巴拉奥（Subbarao）猜想。据我们所知，这是文献中对苏巴拉奥猜想的首次推广。
