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.