关于第二类Stirling数的一个恒等式的三种证明

Three Proofs of an Identity Involving Stirling Numbers of the Second Kind

摘要: Sheila Sundaram在研究对称群上关于子词序的具有特定秩的子偏序集的同调表示时，得到了一个关于第二类Stirling数的恒等式，并提出如何给出此恒等式的一个组合证明这样一个公开问题。本文旨在给出此恒等式的两个新的证明以及重新构造前人使用的一个反号对合以给出一个对合证明，从而回答了Sundaram提出的问题。此外，我们还给出了此恒等式左侧和式的一个组合解释，这一组合解释源自于Mansour和Munagi的结果。

Abstract: Sheila Sundaram obtained an identity between Stirling numbers of the second kind while studying representations of the symmetric group on the homology of rank-selected subposets of subword order. She posed an open question that how to give a combinatorial proof of this identity. The aim of the paper is to present two new proofs as well as reproduce a sign-reversing involution proof of this curious identity, thereby answering the question posed by Sundaram. Moreover, we also provide a combinatorial interpretation of the left-hand side of this identity which is originally due to Mansour and Munagi.