  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.
