A generalization of balanced tableaux and marriage problems with unique solutions

Brian Tianyao Chan

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.

Keywords


Balanced tableaux, Hall's marriage condition, shelling

Full Text:

MANUSCRIPT


DOI: https://doi.org/10.26493/1855-3974.2260.c0e

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications