A generalization of balanced tableaux and marriage problems with unique solutions

TL;DR

This paper generalizes balanced tableaux and characterizes shellable set families with unique solutions to marriage problems, providing a new combinatorial framework and a hook-length formula extension.

Contribution

It introduces a natural generalization of balanced tableaux involving set families and words, characterizes shellable families through a strong existence condition, and extends the hook-length formula.

Findings

Shellable families characterized by a strong existence condition.

Average number of generalized tableaux given by a hook-length formula extension.

Provides a unifying framework for marriage problems and balanced tableaux generalizations.

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 $[n]^m$, 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.