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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)!)].