The Ramanujan Journal ( IF 0.6 ) Pub Date : 2020-09-18 , DOI: 10.1007/s11139-020-00315-5 Bernard L. S. Lin , Lin Peng , Pee Choon Toh
Stanley introduced a partition statistic \(srank (\pi )=\mathcal {O}(\pi )-\mathcal {O}(\pi ')\), where \(\mathcal {O}(\pi )\) denote the number of odd parts of the partition \(\pi \), and \(\pi '\) is the conjugate of \(\pi \). Let \(p_i(n)\) denote the number of partitions of n with srank \(\equiv i\pmod 4\). Andrews proved the following refinement of Ramanujan’s partition congruence modulo 5:
$$\begin{aligned} p_0(5n+4)\equiv p_2(5n+4)\equiv 0\pmod 5. \end{aligned}$$In this paper, we consider an analogous partition statistic
$$\begin{aligned} lrank (\pi )=\mathcal {O}(\pi )+\mathcal {O}(\pi '). \end{aligned}$$Let \(p_i^+(n)\) denote the number of partitions of n with lrank \(\equiv i \pmod 4\). We will establish the generating functions of \(p_0^+(n)\) and \(p_2^+(n)\) and show that they satisfy similar properties to \(p_i(n)\). We also utilize a pair of interesting q-series identities to obtain a direct proof of the congruences
$$\begin{aligned} p_0^+(5n+4)\equiv p_2^+(5n+4)\equiv 0\pmod 5. \end{aligned}$$中文翻译：
Andrews-Stanley分区函数的变体和两个有趣的q系列恒等式
Stanley引入了分区统计\（srank（\ pi）= \ mathcal {O}（\ pi）-\ mathcal {O}（\ pi'）\），其中\（\ mathcal {O}（\ pi）\）表示分区\（\ pi \）的奇数部分，而\（\ pi'\）是\（\ pi \）的共轭数。令\（p_i（n）\）表示n的分区数，且顺序为\（\ equiv i \ pmod 4 \）。安德鲁斯证明了Ramanujan的分区同余模5的以下改进：
$$ \ begin {aligned} p_0（5n + 4）\ equiv p_2（5n + 4）\ equiv 0 \ pmod5。\ end {aligned} $$在本文中，我们考虑了类似的分区统计
$$ \ begin {aligned} lrank（\ pi）= \ mathcal {O}（\ pi）+ \ mathcal {O}（\ pi'）。\ end {aligned} $$令\（p_i ^ +（n）\）用lrank \（\ equiv i \ pmod 4 \）表示n的分区数。我们将建立\（p_0 ^ +（n）\）和\（p_2 ^ +（n）\）的生成函数，并证明它们满足与\（p_i（n）\）类似的特性。我们还利用一对有趣的q系列恒等式来获得全等的直接证明。
$$ \ begin {aligned} p_0 ^ +（5n + 4）\ equiv p_2 ^ +（5n + 4）\ equiv 0 \ pmod 5 \ end {aligned} $$
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