The walled Brauer algebra \mathfrak{B}_{r,s}(\delta) is a subalgebra of \mathfrak{B}_{r+s}(\delta). Diagrammatically, it consists of diagrams where the only allowed pairings are of the types X_{i\leq r}-X_{j>r}, Y_{i\leq r}-Y_{j>r}, X_{i\leq r}-Y_{j\leq r}, X_{i>r} - Y_{j>r}. This amounts to having a wall that separates X_{i\leq r},Y_{i\leq r} from X_{i>r},Y_{i>r}, and requiring that X-Y pairings cross the wall while X-X,Y-Y pairings don't.
