The center of the walled Brauer algebra $B_{r,1}(\delta )$

We show that the centre of the walled Brauer algebra $B_{r,1}(\delta )$ over the complex field $\mathbb{C}$, for any parameter $\delta \in \mathbb{C}$, is generated by the supersymmetric polynomials evaluated at the Jucys-Murphy elements. Moreover, we prove that its dimension is independent of the parameter $\delta$.