Gr\"obner bases of ideals cogenerated by Pfaffians

We characterise the class of one-cogenerated Pfaffian ideals whose natural generators form a Gr"obner basis with respect to any anti-diagonal term-order. We describe their initial ideals as well as the associated simplicial complexes, which turn out to be shellable and thus Cohen-Macaulay. We also provide a formula for computing their multiplicity.