Given kernel function $k(x,y)=2PQ$, where $P=| \{ (i,j); 1 \le i \lt j \le n, (x_{i} - x_{j})(y_{i}-y_{j}) \ge 0 \}|$ and $Q=| \{ (i,j); 1 \le i \lt j \le n, (x_{i} - x_{j})(y_{i}-y_{j}) \lt 0 \}|$.
Find $\phi$ such that $k(x,y) = 2PQ = \langle\phi (x), \phi (y)\rangle$.
I don't know how to proceed. The scalar product gives a sum of terms, but I cannot find a way to turn it into a product in order to obtain $k$.
Any advice?