A generalization of balanced tableaux and marriage problems with unique solutions
Abstract
We consider families of finite sets that we call shellable and that have been characterized by Chang and by Hirst and Hughes as being the families of sets that admit unique solutions to Hall's marriage problem. In this paper, we introduce a natural generalization of Edelman and Greene's balanced tableaux that involves families of sets that satisfy Hall's marriage Condition and certain words in , then prove that shellable families can be characterized by a strong existence condition relating to this generalization. As a consequence of this characterization, we show that the average number of such generalized tableaux is given by a generalization of the hook-length formula.
Keywords
Cite
@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}
}
Comments
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