Abstract We give a bijection between the set of ordinary partitions and that of self-conjugate partitions with some restrictions. Also, we show the relationship between hook lengths of a self-conjugate partition and its corresponding partition via the bijection. As a corollary, we give new combinatorial interpretations for the Catalan number and the Motzkin number in terms of self-conjugate simultaneous core partitions.

中文翻译：

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具有相同视差的自共轭分区

摘要 我们给出了具有一些限制的普通分区集和自共轭分区集之间的双射。此外，我们通过双射展示了自共轭分区的钩子长度与其对应分区之间的关系。作为推论，我们根据自共轭同时核心分区对 Catalan 数和 Motzkin 数给出了新的组合解释。