Decreasing subsequences and Viennot for oscillating tableaux

We establish an extension of Viennot’s geometric (shadow line) construction to the setting of oscillating tableaux. We then use this to give a new proof of the Type C analogue of Schensted’s theorem on longest decreasing subsequences. This pairs with our results from [3] on Type C webs to 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.