  We give a direct proof of a result of Sundaram and Stanley: that the
dimension of the space of invariant vectors in a $2k$-fold tensor product of
the vector representation of $\mathfrak{sp}_{2n}$ equals the number of
$(n+1)$-avoiding matchings of $2k$ points. This can be viewed as an extension
of Schensted's theorem on longest decreasing subsequences. Our main tool is an
extension of Viennot's geometric construction to the setting of up-down
tableaux.
