论文标题
Glaisher地图和Sylvester定理的概括的某些后果
Some consequences of Glaisher's map and a generalization of Sylvester's theorem
论文作者
论文摘要
对于正整数$ k，l \ geq 2 $，$ k $的零件套件最多出现$ l $ times的零件引起了很多兴趣，因为可以使用Glaisher的映射组成来证明某些情况下的相关分区身份。我们考虑一些特殊情况并得出一些算术特性。特别的重点是零件奇怪且独特的分区集（$ k = 2 $，$ l = 2 $）。西尔维斯特（Sylvester）证明，对于固定重量，这组分区与一组自轭分区相同。我们介绍了一类新的分区，这些分区概括了自轭分区，因此，我们扩展了Sylvester的定理。此外，使用此类分区，我们对A. K. Agarwal先前考虑的一些Rogers-Ramanujan身份提供了新的组合解释。
For positive integers $k, l \geq 2$, the set of $k$-regular partitions in which parts appear at most $l$ times has attracted a lot of interest in that a composition of Glaisher's mapping can be used to prove the associated partition identities in certain cases. We consider some special cases and derive some arithmetic properties. Of particular focus is the set of partitions in which parts are odd and distinct ($k =2$, $l = 2$). Sylvester proved that, for fixed weight, this set of partitions is equinumerous with the set of self-conjugate partitions. We introduce a new class of partitions that generalizes self-conjugate partitions and as a result, we extend Sylvester's theorem. Furthermore, using this class of partitions, we give new combinatorial interpretation of some Rogers-Ramanujan identities which were previously considered by A. K. Agarwal.
