Abstract Recently, Andrews defined a partition function EO ( n ) which counts the number of partitions of n in which every even part is less than each odd part. He also defined a partition function EO ‾ ( n ) which counts the number of partitions of n enumerated by EO ( n ) in which only the largest even part appears an odd number of times. Andrews proposed to undertake a more extensive investigation of the properties of EO ‾ ( n ) . In this article, we prove infinite families of congruences for EO ‾ ( n ) . We next study distribution of EO ‾ ( n ) . We prove that there are infinitely many integers N in every arithmetic progression for which EO ‾ ( 2 N ) is even; and that there are infinitely many integers M in every arithmetic progression for which EO ‾ ( 2 M ) is odd so long as there is at least one. We further prove that EO ‾ ( n ) is even for almost all n. Very recently, Uncu has treated a different subset of the partitions enumerated by EO ( n ) . We prove that Uncu's partition function is divisible by 2 k for almost all k. We use arithmetic properties of modular forms and Hecke eigenforms to prove our results.

中文翻译：

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关于偶数部分低于奇数部分的安德鲁斯整数分区

摘要 最近，Andrews 定义了一个分区函数 EO ( n )，它计算 n 中每个偶数部分都小于每个奇数部分的分区数。他还定义了一个分区函数 EO ‾ ( n )，它计算 EO ( n ) 枚举的 n 的分区数，其中只有最大的偶数部分出现奇数次。Andrews 提议对 EO ‾ (n) 的性质进行更广泛的研究。在本文中，我们证明了 EO ‾ ( n ) 的无限同余族。我们接下来研究 EO ‾ ( n ) 的分布。我们证明了在每个等差数列中存在无穷多个整数 N，其中 EO ‾ ( 2 N ) 为偶数；并且在每个等差数列中都有无穷多个整数 M，其中 EO ‾ ( 2 M ) 是奇数，只要至少有一个。我们进一步证明 EO ‾ ( n ) 对于几乎所有的 n 都是偶数。最近，Uncu 处理了由 EO ( n ) 枚举的分区的不同子集。我们证明，对于几乎所有的 k，Uncu 的分区函数都可以被 2 k 整除。我们使用模形式和 Hecke 特征形式的算术性质来证明我们的结果。