Arithmetic properties of partitions with designated summands

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• 作者：Ernest X.W. Xia ^{Ernestxwxia@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11P83 ; 05A17
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：159
• 期：Complete
• 页码：160-175
• 全文大小：316 K

文摘

A new class of partitions, partitions with designated summands, was introduced by Andrews, Lewis and Lovejoy. Let e64547d495ce7a391" title="Click to view the MathML source">PD(n)$\mathit{PD}(n)$ denote the number of partitions of n   with designated summands. Andrews, Lewis and Lovejoy established many congruences modulo 3 and powers of 2 for e64547d495ce7a391" title="Click to view the MathML source">PD(n)$\mathit{PD}(n)$ by using the theory of modular forms. In this paper, we prove several infinite families of congruences modulo 9 and 27 for e64547d495ce7a391" title="Click to view the MathML source">PD(n)$\mathit{PD}(n)$ by employing the generating functions of PD(3n)$\mathit{PD}(3n)$ and PD(3n+1)$\mathit{PD}(3n+1)$ which were discovered by Chen, Ji, Jin and Shen. For example, we prove that for e62883d5bc5921f92de6ffae358e" title="Click to view the MathML source">n≥0$n\ge 0$ and e639e59033edb2058424b" title="Click to view the MathML source">k≥1$k\ge 1$, e699784c019f9ac7c">$\mathit{PD}(2^{18k-1}(12n+1))\equiv 0(\mathrm{mod}27)$. Furthermore, using some results due to Newman, we find some strange congruences modulo 27 for e64547d495ce7a391" title="Click to view the MathML source">PD(n)$\mathit{PD}(n)$. For example, we prove that for k≥0$k\ge 0$, $\mathit{PD}({13}^{9k}(75p+2))\equiv 0(\mathrm{mod}27)$ and $\mathit{PD}(2\times {13}^{9k+8})\equiv 0(\mathrm{mod}27)$, where p   is a prime and $p\equiv 1(\mathrm{mod}12)$.