Involution with exactly one fixed point ($f^2=\mathrm{id},|\mathrm{Fix}(f)|=1$)

An involution partitions its set into fixed points and pairs exchanged by the function. If there is exactly one fixed point, every finite model has odd size. Conversely any finite set of odd size admits such a function by pairing all but one element. The first-order sentence $\forall xf(f(x))=x\wedge \exists z(f(z)=z\wedge \forall x(f(x)=x\to x=z))$ consequently has exactly the positive odd finite model sizes. Its arbitrarily large finite models imply an infinite model by the compactness theorem.