Counting Lattice Paths By Gessel Pairs
  We count a large class of lattice paths by using factorizations of free monoids. Besides the classical lattice paths counting problems related to Catalan numbers, we give a new approach to the problem of counting walks on the slit plane (walks avoid a half line) that was first solved by Bousquet-M\'{e}lou and Schaeffer. We also solve a problem about walks in the half plane avoiding a half line by subsequently applying the factorizations of two different Gessel pairs, giving a generalization of a result of Bousquet-M\'{e}lou. 
Guoce Xin