In this paper, we develop a new method to compute generating functions of the form NMΟβ(t,x,y)=nβ₯0βββtnβn!βΟβlNMnβ(Ο)βxLRMin(Ο)y1+des(Ο) where Ο is a permutation that starts with 1,NMnβ(Ο) is the set of permutations in the symmetric group Snβ with no Ο -matches, and for any permutation ΟβSnβ, LRMin(Ο) is the number of left-to-right minima of Ο and des(Ο) is the number of descents of Ο . Our method does not compute NMΟβ(t,x,y) directly, but assumes that NMΟβ(t,x,y)=/β(UΟβ(t,y))x1β where UΟβ(t,y)=βnβ₯0βUΟβ,n(y)βn!tnβ so that UΟβ(t,y)=βNMΟβ(t,1,y)1β. We then use the so-called homomorphism method and the combinatorial interpretation of NMΟβ(t,1,y) to develop recursions for the coefficient of UΟβ(t,y)