A generalization of balanced tableaux and marriage problems with unique solutions

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