Séminaire Lotharingien de Combinatoire, B59f (2008), 13 pp.

Jean-Christophe Aval

Keys and Alternating Sign Matrices

Abstract. 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.

Received: March 13, 2008. Accepted: October 4, 2008. Final Version: October 16, 2008.

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Erratum by Jean-Christophe Aval

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

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