Geodesic
A generalization of balanced tableaux and marriage problems with unique solutions
Ars Mathematica Contemporanea, Tome 21 (2021) no. 2, article no. 03, 17 p.

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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.
DOI : 10.26493/1855-3974.2260.c0e
Keywords: Balanced tableaux, Hall's marriage condition, shelling 
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Brian Tianyao Chan. A generalization of balanced tableaux and marriage problems with unique solutions. Ars Mathematica Contemporanea, Tome 21 (2021) no. 2, article  no. 03, 17 p. doi : 10.26493/1855-3974.2260.c0e. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.2260.c0e/

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