Geodesic
Keys and Alternating Sign Matrices
Séminaire lotharingien de combinatoire, Tome 59 (2008-2010)

Voir la notice de l'acte provenant de la source Séminaire Lotharingien de Combinatoire website

In [Invariant Theory and Tableaux, I.M.A. Vol. Math. Appl. 19, Springer-Verlag, New York, 1990, pp. 125-144], Lascoux and Schützenberger introduced a notion of key associated to any Young tableau. More recently, Lascoux defined the key of an alternating sign matrix by recursively removing all -1's in such matrices. But alternating sign matrices are in bijection with monotone triangles, which form a subclass of Young tableaux. We show that in this case these two notions of keys coincide. Moreover we obtain an elegant and direct way to compute the key of any Young tableau, and discuss consequences of our result.

Erratum by Jean-Christophe Aval

As Florent le Gac points out, the formula giving An(2) at the bottom of page 11 contains an error. The correct formula is:

An=(n!)2[1/(2592*(n-6)!)+11/(3600*(n-5)!)+1/(288*(n-4)!)]. 

@article{SLC_2008-2010_59_a4,
     author = {Jean-Christophe Aval},
     title = {Keys and {Alternating} {Sign} {Matrices}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {59},
     year = {2008-2010},
     url = {http://geodesic.mathdoc.fr/item/SLC_2008-2010_59_a4/}
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Jean-Christophe Aval. Keys and Alternating Sign Matrices. Séminaire lotharingien de combinatoire, Tome 59 (2008-2010). http://geodesic.mathdoc.fr/item/SLC_2008-2010_59_a4/