论文标题
贝克型分区身份的改进
Refinements of Beck-type partition identities
论文作者
论文摘要
富兰克林（Franklin）的身份概括了欧拉（Euler）的身份，并指出，$ n $的分区数量，$ j $不同的零件可除以$ r $，等于$ n $的分区数量和$ j $重复的零件。在本文中，当$ j = 1 $时，我们将对富兰克林的身份进行改进。当$ j = 1 $，$ r = 2 $的分区（即固定周长，即固定最大的钩子）时，我们证明了富兰克林的身份。我们还为具有固定周边的分区提供了一个贝克型身份：所有分区中的零件数量的过剩零件的过量零件$ m $比所有分区中的零件数量均超过所有分区的零件数量，而零件的零件数则在不同的零件中，外围$ m $等于零件$ m $，其零件均为单件零件。我们提供结果的分析和组合证明。
Franklin's identity generalizes Euler's identity and states that the number of partitions of $n$ with $j$ different parts divisible by $r$ equals the number of partitions of $n$ with $j$ repeated parts. In this article, we give a refinement of Franklin's identity when $j=1$. We prove Franklin's identity when $j=1$, $r=2$ for partitions with fixed perimeter, i.e., fixed largest hook. We also derive a Beck-type identity for partitions with fixed perimeter: the excess in the number of parts in all partitions into odd parts with perimeter $M$ over the number of parts in all partitions into distinct parts with perimeter $M$ equals the number of partitions with perimeter $M$ whose set of even parts is a singleton. We provide analytic and combinatorial proofs of our results.
