平衡表状的推广与具有唯一解的婚姻问题
组合数学
2021-10-04 v2
摘要
我们考虑一类称为可壳分解（shellable）的有限集族，Chang 以及 Hirst 和 Hughes 已刻画其为满足霍尔婚姻问题存在唯一解的集族。本文中，我们引入 Edelman 与 Greene 平衡表状的一个自然推广，其涉及满足霍尔婚姻条件（Hall's marriage Condition）的集族以及 中的特定字，进而证明可壳分解集族可由关联于该推广的一个强存在性条件来刻画。作为该刻画的一个推论，我们证明此类广义表状的平均数目由一个钩长公式的推广给出。
引用
@article{arxiv.1909.05324,
title = {A generalization of balanced tableaux and marriage problems with unique solutions},
author = {Brian Chan},
journal= {arXiv preprint arXiv:1909.05324},
year = {2021}
}
备注
Main result has been generalized to certain words in $[n]^m$. We thank an anonymous referee for finding a gap in the first half of the proof of Theorem 3.14 in the old draft, which we have fixed. Moreover, to Lemma 3.7 in the old draft, we have added a necessary extra assumption