A generalization of balanced tableaux and marriage problems with unique solutions - Chan, Brian Tianyao (Ars mathematica contemporanea, 2021)
We consider families of finite sets that we call flagged and that have been characterized by Chang as being the families of sets that admit unique solutions to Hall’s marriage problem and we consider generalizations of Edelman and Greene’s balanced tableaux previously investigated by Viard. 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 ?$m^n$?, then prove that flagged families can be characterized by a strong existence condition relating to this generalization. As a consequence of this characterization, we show that the arithmetic mean of the sizes of subclasses of such generalized tableaux is given by a generalization of the hook-length formula.
Keywords: balanced tableaux; granatiranje; Hallov pogoj v zvezi s problemom porok; non-CayleyHall’s marriage condition; ravnotežene tablice; shelling