the partial fraction algorithm of guoce xin has recently led to a breakthrough in the theory of tesler matrices. in particular we now have a beautiful formula for the polynomials enumerating the families of tesler matrices with positive hook weights. the xin algorithm also yields a very illuminating new proof of the original tesler matrix formula for the hilbert series of diagonal harmonics due to jim haglund. we will try to give a glimpse of these developments in a self contained manner.