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 arXiv:2103.14997v1 [math.RT] 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.

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